open

en-adj

open (comparative more open, superlative most open)

more open (comparative)
most open (superlative)

closed-source

antonym(s) of “with a free license and no proprietary components”

From Middle English open, from Old English open (“open”), from Proto-West Germanic *opan, from Proto-Germanic *upanaz (“open”), from Proto-Indo-European *upo (“up from under, over”).
cognates
* Scots apen (“open”)
* Saterland Frisian eepen (“open”)
* West Frisian iepen (“open”)
* Dutch open (“open”)
* Low German open, apen (“open”)
* German offen (“open”)
* Danish åben (“open”)
* Swedish öppen (“open”)
* Norwegian Bokmål åpen (“open”)
* Norwegian Nynorsk open (“open”)
* Icelandic opinn (“open”)
 Compare also Latin supinus (“on one's back, supine”), Albanian hap (“to open”). Related to up.

der

at church every time the doors swing open

(mathematics, topology, of a set) Which is part of a predefined collection of subsets of X, that defines a topological space on X.

obert

math: which is part of a predefined collection of subsets


chain

From Middle English cheyne, chaine, from Old French chaine, chaene (“chain”), from Latin catēna (“chain”), from Proto-Indo-European *kat- (“to braid, twist; hut, shed”). Doublet of catena.

a chain is only as strong as its weakest link

(algebraic topology, originally) A formal sum of cells in a CW complex of a certain dimension k (in which case the formal sums are called k'''-chains); a formal sum of simplices or cubes of a certain dimension in a simplical complex or cubical complex (respectively).

(algebraic topology, homological algebra, more generally) An element of a group (or module) in a chain complex.


path

From Middle English path, peth, from Old English pæþ (“path, track”), from Proto-West Germanic *paþ, from Proto-Germanic *paþaz (“path”). The Proto-Germanic term could be borrowed from Iranian, from Proto-Iranian *pántaHh, from Proto-Indo-Iranian *pántaHs, from Proto-Indo-European *póntoh₁s, from the root *pent- (“to pass”), but many go against it and it is far from being universally accepted. https://blog.oup.com/2015/11/path-word-origin-etymology/
Germanic cognates include West Frisian paad, Dutch pad, German Pfad. Indo-Iranian cognates could be Avestan 𐬞𐬀𐬧𐬙𐬃 (paṇtā̊, “way”), Old Persian 𐎱𐎰 (p-θ /paθi/)), Sanskrit पन्था (panthā). See also English find). Doublet of panth.

(topology) A continuous map f from the unit interval I=[0,1] to a topological space X.


braid

en-noun

braid (countable and uncountable, plural braids)

From Middle English braiden, breided, bræiden, from Old English breġdan (“to move quickly, pull, shake, swing, throw (wrestling), draw (sword), drag; bend, weave, braid, knit, join together; change color, vary, be transformed; bind, knot; move, be pulled; flash”), from Proto-West Germanic *bregdan, from Proto-Germanic *bregdaną (“to flicker, flutter, jerk, tug, twitch, flinch, move, swing”), from Proto-Indo-European *bʰrēḱ-, *bʰrēǵ- (“to shine, shimmer”).
Cognate with Scots Scots brade, Scots braid (“to move quickly or suddenly”), Saterland Frisian braidje (“to knit”), West Frisian breidzje, Dutch breien (“to knit”), Low German breiden, German breiden, Bavarian bretten (“to move quickly, twitch”), Icelandic bregða (“to move quickly, jerk”), Faroese bregða (“to move quickly, react swiftly; to draw (sword)”) and Faroese bregda (“to plaid, braid, twist, twine”).

cog

We introduce braids via their historical roots and uses, make connections with knot theory and present the mathematical theory of braids through the braid group.

2009, Mitchell A. Berger, Louis H. Kauffman, Renzo L. Ricca, Lectures on Topological Fluid Mechanics, page 1:

In order to characterise the structure and complexity of a braid different numbers or topological invariants can be calculated.

2012, A. T. Skjeltorp, Tamas Vicsek, Complexity from Microscopic to Macroscopic Scales, page 144:

(mathematics, topology) Given two sets of n points on corresponding positions on two parallel lines, a braid is a unique set of crossings (over or under) between n strands that connect each point on one line to a point on the other line such that all points represent the terminus of one and only one strand and the traversal of any strand from a starting point to an ending point never moves further away from the from the ending point.


interior

/ɪnˈtɪə.ɹɪ.ə/

From Latin interior (“inner, interior”).

(mathematics, topology) The set of all interior points of a set.


weight

en-noun

weight (countable and uncountable, plural weights)

Belizean Creole: wayt

→ Japanese: ウエイト (ueito)

→ Burmese: ဝိတ် (wit)

From Middle English weight, weiȝte, weght, wight, from Old English wiht, ġewiht (“weight”), from Proto-Germanic *wihtiz ("weight"; compare *weganą (“to move”)), from Proto-Indo-European *weǵʰ- (“to move; pull; draw; drive”). Equivalent to weigh + -th.
Cognate with Scots wecht, weicht (“weight”), Saterland Frisian Wächte (“scale”), Gewicht (“weight”), West Frisian gewicht (“weight”), Dutch gewicht (“weight”), German Low German Wicht, Gewicht (“weight”), German Wucht (“massiveness, force”), Gewicht (“weight”).

gloss

"weight"; compare <i class="Latn mention" lang="gem-pro">*weganą</i> (“to move”)

inh

(topology) The smallest cardinality of a base.

paino

topology: the smallest cardinality of a base


curve

Attested since the 1690s, from Latin curvus (“bent, curved”), ultimately from Proto-Indo-European *(s)ker- (“to bend, curve, turn”) + *-wós. Doublet of curb, shrink, carcer, and cancer.


basis

en-noun

basis (plural bases or (rare) baseis or (nonstandard) basises)

bases (plural)
baseis (plural rare)
basises (nonstandard plural)

From Latin basis, from Ancient Greek βάσις (básis), from Proto-Indo-European *gʷémtis, derived from Proto-Indo-European *gʷem- (whence also come). Doublet of base.

The collection of all possible unions of basis elements of a basis is said to be the topology generated by that basis.

(topology) A collection of subsets ("basis elements") of a set, such that this collection covers the set, and for any two basis elements which both contain an element of the set, there is a third basis element contained in the intersection of the first two, which also contains that element.


cover

cover (countable and uncountable, plural covers)

→ Hijazi Arabic: كَڤَر (kavar)

→ Portuguese: cover

Etymology tree
Proto-Indo-European *ḱe-?
Proto-Indo-European *ḱóm
Proto-Italic *kom
Old Latin com
Latin cum
Latin con-
Proto-Indo-European *h₁ep-?
Proto-Indo-European *h₁epi
Proto-Indo-European *h₂wer-
Proto-Indo-European *-yeti
Latin operiō
Latin cooperiō
Old French covrirbor.
Middle English coveren
English cover
From Middle English coveren, borrowed from Old French covrir, cueuvrir (modern French couvrir), from Late Latin coperire, from Latin cooperiō (“I cover completely”), from co- (intensive prefix) + operiō (“I close, cover”). Displaced native Middle English thecchen and bethecchen (“to cover”) (from Old English þeccan, beþeccan (“to cover”)), Middle English helen, (over)helen, (for)helen (“to cover, conceal”) (from Old English helan (“to conceal, cover, hide”)), Middle English wrien, (be)wreon (“to cover”) (from Old English (be)wrēon (“to cover”)), Middle English hodren, hothren (“to cover up”) (from Low German hudren (“to cover up”)).
According to the Oxford English Dictionary, the original sense of the verb and noun cover was “hide from view” as in its cognate covert. Except in the limited sense of “cover again”, the word recover is unrelated and is cognate with recuperate. Cognate with Spanish cubrir (“to cover”).

cumhdach

A cloth or similar material used to protect underlying items from dust, sunlight, insects etc. when not being used.

throw nickels around like manhole covers

The open intervals are a cover for the real numbers.

(combinatorics, topology) A collection (or family) of subsets of a given set, whose union contains every element of said original set.


base

en-noun

base (countable and uncountable, plural bases)

acid

antonym(s) of “chemical compound that will neutralize an alkali”

From Middle English base, bas, baas, from Old French base, from Latin basis, from Ancient Greek βάσις (básis). Doublet of basis and bass.

أَسَاس

something from which other things extend

(topology) The set of sets from which a topology is generated.

kanta

topology: set of sets from which a topology is derived

(topology) A topological space, looked at in relation to one of its covering spaces, fibrations, or bundles.

kanta

topology: topological space, looked at in relation to one of its covering spaces, fibrations, or bundles


handle

desctree

→ Japanese: ハンドル (handoru)
→ Hokkien: 捍多路 (hăn-tó͘-luh)
→ Sakizaya: handulu

From Middle English handel, handle, from Old English handle (“a handle”), from handlian (“to handle, feel, deal with, discuss”). See verb below. Cognate with Danish handel (“a handle”).

Such a 2-handle cancels the 1-handle so the manifold is D⁴.

2003, Gordana Matić, Clint McCrory, Topology and geometry of manifolds, →ISBN, page 182:

(topology) A topological space homeomorphic to a ball but viewed as a product of two lower-dimensional balls.


domain

range

antonym(s) of “domain of definition of a function”

From Middle English demayne, demain (“rule”), from Old French demeine, demaine, demeigne, domaine (“power”), (French domaine), from Latin dominium (“property, right of ownership”), from dominus (“master, proprietor, owner”). See dame, and compare demain, danger, dungeon. Doublet of demesne, domino.

definisieversameling

(mathematics) the set on which a function is defined

mathematical analysis
mathematical analysis

(mathematics, topology, mathematical analysis) An open and connected set in some topology. For example, the interval (0,1) as a subset of the real numbers.


suspension

en-noun

suspension (countable and uncountable, plural suspensions)

Borrowed from Late Latin suspensiōnem (“arching, vaulting; suspension”), from suspendēre (“to hang up, to suspend”), from sub- (“under”) + pendere (“to hang, to suspend”), from Proto-Italic *pendō (“to hang, to put in a hanging position”), from Proto-Indo-European *(s)pénd-e-ti, from *(s)pend- (“to pull; to spin”)). Compare Anglo-Norman suspensiun, French suspension, Occitan suspensio.

der

To get an intuitive feeling for the characteristics of H'-spaces, it is instructive to consider an important class of such spaces, the suspensions. The suspension of an arbitrary topological space Y is defined to be the quotient space of Y#x5C;timesI where Y#x5C;times 0 is identified to one point and Y#x5C;times 1 is identified to another point. For example, the suspension of a circle is a cylinder with the two ends collapsed into one point each; in other words, a space homeomorphic to a sphere.

2012, H. Rasmussen, “Strategy-proofness of Continuous Aggregation Maps”, in Geoffrey M. Heal, editor, Topological Social Choice: With 45 Figures (Social Choice and Welfare; vol. 14, no. 2, 1997), Berlin: Springer-Verlag, →DOI, →ISBN, page 110:

(topology) A topological space derived from another by taking the product of the original space with an interval and collapsing each end of the product to a point.

suspensie

topology: A topological space derived from another by taking the product of the original space with an interval and collapsing each end of the product to a point

A model category is called pointed if the initial object and terminal object are the same. The homotopy category of any pointed model category acquires a suspension functor denoted by #x5C;Sigma. It turns out that #x5C;text#x7B;Ho#x7D;(M) is a pre-triangulated category in a natural way[…]. When the suspension is an equivalence, M is called a stable model category, and in this case #x5C;text#x7B;Ho#x7D;(M) becomes a triangulated category[…].

2010, Paul Arne Østvær, “Preliminaries”, in Homotopy Theory of C*-Algebras (Frontiers in Mathematics), Basel: Birkhäuser, Springer Basel, →DOI, →ISBN, page 17:

(topology) A function derived, in a standard way, from another, such that the instant function’s domain and codomain are suspensions of the original function’s.

suspensie

topology: A function derived, in a standard way, from another, such that the instant function’s domain and codomain are suspensions of the original function’s


cycle

→ Japanese: サイクル (saikuru)

From Middle English cicle (“fixed length period of years”), from Late Latin cyclus, from Ancient Greek κύκλος (kúklos, “circle”), from Proto-Hellenic *kúklos, *kʷókʷlos, from Proto-Indo-European *kʷékʷlos (“circle, wheel”).
Doublet of chakra, chakram, charkha, chukker, cyclus, and wheel (see there for more).

pyörä

bicycle, tricycle, motorcycle etc. — see also bicycle

(topology, algebraic topology) A chain whose boundary is zero.


discrete

en-adj

discrete (comparative more discrete, superlative most discrete)

more discrete (comparative)
most discrete (superlative)

From Old French discret, from Latin discrētus, past participle of discernō (“divide”), from dis- + cernō (“sift”). Doublet of discreet.

(topology) Having each singleton subset open: said of a topological space or a topology.


graph

Clipping of graphic formula. From 1878; verb from 1889.

2008, Unnamed translators (AMS), A. V. Alexeevski, S. M. Natanzon, Hurwitz Numbers for Regular Coverings of Surfaces by Seamed Surfaces and Cardy-Frobenius Algebras of Finite Groups, V. M. Buchstaber, I. M. Krichever (editors), Geometry, Topology, and Mathematical Physics: S.P. Novikov's Seminar, 2006-2007, American Mathematical Society, page 6,
First, let us define its 1-dimensional analog, that is, a topological graph. A graph Δ is a 1-dimensional stratified topological space with finitely many 0-strata (vertices) and finitely many 1-strata (edges). […] A graph such that any vertex belongs to at least two half-edges we call an s-graph. Clearly the boundary ∂Ω of a surface Ω with marked points is an s-graph.
A morphism of graphs φ:Δ'→Δ is a continuous epimorphic map of graphs compatible with the stratification; i.e., the restriction of φ to any open 1-stratum (interior of an edge) of Δ' is a local (therefore, global) homeomorphism with appropriate open 1-stratum of Δ.

(topology) A topological space which represents some graph (ordered pair of sets) and which is constructed by representing the vertices as points and the edges as copies of the real interval [0,1] (where, for any given edge, 0 and 1 are identified with the points representing the two vertices) and equipping the result with a particular topology called the graph topology.


genus

en-noun

genus (plural genera or (all nonstandard) genuses or genusses or genii)

genera (plural)
genuses (nonstandard plural)
genusses (nonstandard plural)
genii (nonstandard plural)

→ Malayalam: ജീനസ് (jīnasŭ)

→ Swahili: jenasi

→ Thai: สกุล (sà-gun) (semantic loan)

Etymology tree
Proto-Indo-European *ǵenh₁-
Proto-Indo-European *-os
Proto-Indo-European *ǵénh₁os
Latin genusbor.
English genus
Borrowed from Latin genus (“birth, origin, a race, sort, kind”) from the root gen- in Latin gignō (“to beget, produce”). Doublet of gender, genre, and kin.

genus

rank in a taxonomic classification between family and species

(topology, graph theory, algebraic geometry) A natural number representing any of several related measures of the complexity of a given manifold or graph.


infinity

en-noun

infinity (countable and uncountable, plural infinities)

From Middle English infinite, from Old French infinité, from Latin infinitas (“unlimitedness”), from negative prefix in- (“not”), + finis (“end”), + noun of state suffix -tas.

(countable, topology, mathematical analysis) An idealised point which is said to be approached by sequences of values whose magnitudes increase without bound.


function

From Middle French function, from Old French fonction, from Latin functiō (“performance, execution”), from functus, perfect participle of fungor (“to perform, execute, discharge”), from Proto-Indo-European *bʰewg- (“to enjoy”).

دَالَّة

mathematics: a relation between a set of inputs and a set of permissible outputs

(computing) A routine that receives zero or more arguments and may return a result.

وَظِيفَة

computing: routine that returns a result


torus

Borrowed from Latin torus (“a round, swelling, elevation, protuberance”).

bor

A 4-variable Karnaugh map can be thought of, topologically, as being a torus.

(geometry) The standard representation of such a space in 3-dimensional Euclidean space: a surface or solid formed by rotating a closed curve, especially a circle, about a line which lies in the same plane but does not intersect it (e.g. like a ring doughnut).

(topology) A topological space which is a product of two circles.

(topology, in combination, n-torus, 4-torus, etc.) The product of the specified number of circles.


closed

open

antonym(s) of “also phonetics (of vowels, syllables)”

(topology, of a set) Having an open complement.


manifold

From Middle English manifold, from Old English maniġfeald (“manifold, various, varied, complicated, numerous, abundant, plural”), from Proto-Germanic *managafalþaz, equivalent to many + -fold. Cognate with Middle High German manecvalt (“manifold”), Icelandic margfaldr (“multiple”). Compare also German mannigfaltig (“various”), Dutch menigvoudig (“various”), Danish mangefold (“multiple”), Swedish mångfald (“diversity”).

cog

(mathematics) A Hausdorff topological space that looks locally like the "ordinary" Euclidean space ℝⁿ.


atlas

Borrowed from Latin Atlas, from the name of the Ancient Greek mythological figure Ἄτλας (Átlas, “Bearer (of the Heavens)”), from τλῆναι (tlênai, “to suffer”, “to endure”, “to bear”). The sense referring to books of maps comes from the Atlas of Mercator, which he named thus in honor of Atlas, who was supposed to be skillful in astronomy and the doctrine of the sphere. The sense referring to the vertebra reflects that the spine carries the globe of the cranium (the neck carries the head).

differential geometry
differential geometry

(differential geometry, topology) A family of coordinate charts that cover a manifold.

atlas

(differential geometry, topology) family of coordinate maps that cover a manifold


proper

en-adj

proper (comparative more proper, superlative most proper)

more proper (comparative)
most proper (superlative)

From Middle English propre, from Anglo-Norman proper, propre, Old French propre (French: propre), from Latin proprius.

(topology, of a function) Such that the preimage of every compact set is compact.

(topology, of a function) Continuous, mapping closed sets to closed sets, and such that the preimage of every point is compact.


lattice

→ Japanese: ラティス (ratisu)

From Middle English latis, from Middle French lattis (“lathing”), from Old French lattis, from latte (“a lath”), from Frankish *lattā, *lattu (“a lath”), from Proto-Germanic *lattǭ, *lattō, *laþþō (“board; plank; ledge”), from Proto-Indo-European *(s)latn-, *(s)lat- (“beam; log”). Cognate with Old High German latta (“lath”), (German Latte), Old English lætt (“lath”), Middle Low German lāde (“plank, counter, sales counter”), German Laden (“shop”). More at lath.

cog

(mathematics, algebra) In any of several technical senses, a substructure with properties analogous to those the set of points with integer coordinates bears in relation to the full Euclidian space these points are contained in.

(topology, Lie theory, generalizing sense 5.1) A discrete subgroup L of a given locally compact group G whose quotient space G/L has finite invariant measure.

Lie theory; generalizing sense 5.1; Lie theory; generalizing sense 5.1


local

en-adj

local (comparative more local, superlative most local)

more local (comparative)
most local (superlative)

From Middle English local, from Late Latin locālis (“belonging to a place”), possibly also via Old French local; ultimately from Latin locus (“a place”).
The ring-theoretic senses derive from Krull, who first referred to Noetherian commutative rings with a unique maximal ideal as "Stellenring" (Stellen (“place”) + ring) in 1938. The term was inspired by algebraic geometry, where local rings encode information about the behavior of curves (surfaces, etc.) at points; hence, describe "local" behavior.

der

A Hausdorff space satisfying local compactness need not be (globally) compact!

(mathematics, not comparable, of a condition or property) Applying to or satisfied by substructures understood as "near points;" in particular:

(topology) Satisfied by at least one open neighborhood of every point.

(topology) Satisfied by arbitrarily small open neighborhoods of every point.


angelic

en-adj

angelic (comparative more angelic, superlative most angelic)

more angelic (comparative)
most angelic (superlative)

/ænˈd͡ʒɛlɪk/

From Middle English angelik, aungillik, aungellike, (also angellich, aungellich > English angelly), from Old English anġelīċ, engellīċ, englelīċ, coalescing with Old French angélique, from Latin angelicus, from Ancient Greek ἀγγελικός (angelikós, “of or for a messenger”), from ἄγγελος (ángelos, “angel”). Equivalent to angel + -ic. Doublet of angelique (“plant of the genus Angelica”) and angélique (“plucked bowl lute”).

doublet

(topology) A regular Hausdorff space is said to be angelic if the closure of each relatively countably compact set A is compact and the closure consists of the limits of sequences in A.


orbifold

Blend of orbit + manifold. Named by William Thurston.

(topology) A topological space in which every small enough neighborhood is homeomorphic to a quotient of real space by the action of a finite group.


chessboard

(topology) A mathematical construction based on this pattern of squares


knot theory

(mathematics) A branch of topology dealing with knots.


simplex

en-noun

simplex (plural simplexes or simplices or simplicia)

simplexes (plural)
simplices (plural)
simplicia (plural)

Henri Poincaré
Pieter Hendrik Schoute
Simplex (disambiguation)
simplex

From Latin simplex (“simple”).
In the mathematical sense, apparently first used (in German) in 1902, Pieter Hendrik Schoute, Mehrdimensionale Geometrie, where Schoute first suggests the term Simplicissimum, but then from the next page decides to use simplex "for short". In his pioneering works on algebraic topology, Henri Poincaré had previously introduced the concept, but not the actual term simplex.

linearly independent
linearly independent

(geometry, algebraic topology) An analogue in a space of arbitrary dimensionality of the triangle or tetrahedron; the convex hull of n+1 linearly independent points in n-dimensional space; in other words, the triangle, tetrahedron etc., generalized to an arbitrary number of dimensions.

单纯形

an analogue in any dimension of the triangle or tetrahedron


chart

→ Japanese: チャート (chāto)

→ Korean: 차트 (chateu)

→ Swahili: chati

→ Welsh: siart

Borrowed from Middle French charte (“card, map”), from Late Latin charta (“paper, card, map”), Latin charta (“papyrus, writing”), from Ancient Greek χάρτης (khártēs, “papyrus, thin sheet”). See charter, card, carte.

der

coordinate chart
coordinate chart#English

(differential geometry, topology) Synonym of coordinate chart.


boundary

From bound + -ary, Old French, from Latin.

(topology) (of a set) the set of points in the closure of a set S, not belonging to the interior of that set

邊界

(topology) the set of points in the closure, not belonging to the interior


section

From Middle English seccioun, from Old French section, from Latin sectiō (“cutting, cutting off, excision, amputation of diseased parts of the body, etc.”), from sectus, past participle of secāre (“to cut”). More at saw.

разрэ́з

surgery: incision or the act of making an incision

A cross-section (image that shows an object as if cut along a plane).

(topology) A function that generalizes the notion of the graph of a function; formally, a continuous right inverse to the projection map of a fiber bundle.


neighborhood

en-noun

neighborhood (countable and uncountable, plural neighborhoods)

From an alteration of earlier neighborred (“neighborhood”), from Middle English neȝeburredde, neheborreden, equivalent to neighbor + -red; the term being interpreted as neighbor + -hood. For change in suffix (-red to -hood), compare brotherhood.
Cognates
Dutch nabijheid (common in modern language), Dutch naburigheid (uncommon in modern language).

veïnat

topology: set containing an open set which contains some specified point

vizinhança

topology: set containing an open ball which contains a specified point

A set containing an open set which contains some specified point.

Alternatively: An open set which contains some specified point.

A set containing an open ball which contains a specified point.

Alternatively: An open ball which contains some specified point.

(topology) The infinitesimal open set of all points that may be reached directly from a given point.

lähiympäristö

topology: set of all points that may be reached directly from a given point


normal

en-adj

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(topology, of a topology or topological space) In which disjoint closed sets can be separated by disjoint neighborhoods.


domination

en-noun

domination (countable and uncountable, plural dominations)

/ˌdɒməˈneɪʃən/

/ˌdɑːməˈneɪʃən/

From Middle English domynacion, from Old French dominacion, from Latin dominātiō (“rule, dominion”), from dominor (“domineer; rule”); see dominate.

With more than 200 research papers published on the algorithmic complexity of domination and related parameters of graphs, it is difficult to know what to cover in a relatively short chapter on the subject, especially since it could take quite a few pages just to cover the preliminaries of computational complexity and the many algorithm design paradigms that have been applied to domination problems.

1998, Teresa W. Haynes, Stephen T. Hedetniemi, Peter J. Slater, Fundamentals of Domination in Graphs, →ISBN, page 299:


foliation

foliation (countable and uncountable, plural foliations)

/fəʊlɪˈeɪʃn/

From French foliation, from Latin folium (“leaf”).

Historically, the formalism which first arose for the material we discuss is that of measured foliations in surfaces.

1992, R. C. Penner, Combinatorics of Train Tracks, page 204:

We will show that every closed 3-manifold has a foliation of codimension one. In 1952, G. Reeb published his construction of a foliation of the 3-sphere. About twelve years later, W. Lickorish [123] exhibited foliations of codimension one on every closed, orientable 3-manifold.

2003, Alberto Candel, Lawrence Conlon, Foliations, Vol.2, p.253

The simplest example of a foliation is provided by a single submersion F : M → N, M and N being manifolds.

2004, Paweł Grzegorz Walczak, Dynamics Of Foliations, Groups And Pseudogroups, Monografie Matematyczne: Vol.64, New Series, p.6

(topology) A set of submanifolds of a given manifold, each of which is of lower dimension than it, but which, taken together, are coextensive with it.

叶层结构

topology: a set of subspaces coextensive with a manifold


shell

en-verb

shell (third-person singular simple present shells, present participle shelling, simple past and past participle shelled)

shells (present singular third-person)
shelling (participle present)
shelled (participle past)
shelled (past)

From Middle English schelle, from Old English sċiell, from Proto-West Germanic *skallju, from Proto-Germanic *skaljō, from Proto-Indo-European *(s)kelH- (“to split, cleave”).
Compare West Frisian skyl (“peel, rind”), Dutch schil (“peel, skin, rink”), Low German Schell (“shell, scale”), Irish scelec (“pebble”), Latin silex (“pebble, flint”), siliqua (“pod”), Old Church Slavonic сколика (skolika, “shell”). More at shale.
Doublet of sheal.
* (computing): From being viewed as an outer layer of interface between the user and the operating-system internals.


collection

en-noun

collection (countable and uncountable, plural collections)

From Middle English colleccioun, collection, from Old French collection, from Latin collēctiō, collēctiōnem, from collēctus, from colligō (“collect together”), composed of con- + legō (“bring together, gather, collect”), ultimately from Proto-Indo-European *leǵ- (“to gather, collect”).

der

(set theory, topology, mathematical analysis) A set of sets; used because such a thing is in general too large to comply with the formal definition of a set.


dense

en-adj

dense (comparative denser, superlative densest)

denser (comparative)
densest (superlative)

From Middle French dense, from Latin dēnsus, from Proto-Indo-European *dens- (“thick, dense”).

der

(mathematics, topology) Being a subset of a topological space that approximates the space well. See the Wikipedia article on dense sets for a mathematical definition.


excision

en-noun

excision (countable and uncountable, plural excisions)

From Middle French excision, from Latin excīsiō.

(topology) The fact that, under certain hypotheses, the homology of a space relative to a subspace is unchanged by the identification of a subspace of the latter to a point.


cone

From Middle English cone (“corner, angle”) and conoun (“cone”), from Medieval Latin cōnus, cōnon (“cone, wedge, peak”), from Ancient Greek κῶνος (kônos, “cone, spinning top, pine cone”). Reinforced by Middle French cone, from the same Graeco-Latin source.

der

(topology) A space formed by taking the direct product of a given space with a closed interval and identifying all of one end to a point.


vector space

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A vector space is a set of vectors which can be linearly combined.

Each vector space has a basis and dimension.

scalar multiplication
scalar multiplication

(algebra, geometry, topology) A set of elements called vectors, together with some field and operations called addition (mapping two vectors to a vector) and scalar multiplication (mapping a vector and an element in the field to a vector), satisfying a list of constraints; equivalently, a module over a field.


clopen

A subset of a metric space is clopen iff it is both closed and open. A metric space is called zero-dimensional iff there is a base for the open sets consisting of clopen sets.

1990, Gerald A Edgar, Measure, Topology, and Fractal Geometry, page 80:

1999, S. J. Dilworth, On the extensibility of certain homeomorphisms and linear isometries, Krzysztof Jarosz (editor), Function Spaces: Proceedings of the Third Conference on Function Spaces, American Mathematical Society, Contemporary Mathematics, Volume 232, page 124,
FACT 1. Disjoint closed subsets of ℕN may be separated by disjoint clopen sets.

Theorem 3. 1) Every two disjoint closed sets in a totally disconnected compact space #x5C;mathfrak#x7B;Q#x7D; are separated by disjoint clopen sets.
2) Every algebra of clopen sets which separates the points of a totally disconnected compact space contains all clopen sets.

2002, D. A. Vladimirov, Boolean Algebras in Analysis, page 133:

(topology, of a set in a topological space) Both open and closed.


coherent

en-adj

coherent (comparative more coherent, superlative most coherent)

more coherent (comparative)
most coherent (superlative)

/kəʊˈhɪə.ɹənt/

/kəʉˈhɪə.ɹənt/

From Middle French coherent, from Latin cohaērēns, from co- + haereō. By surface analysis, cohere + -ent.

(topology, of a topology 𝒯 on a set T with respect to a family C_a of subsets of T) In a technical sense, determined by the (topology of) the subsets. Formally, Such that 𝒯 is the finest topology on T for which the inclusion maps ιₐ:C_a→T are continuous, where each C_a is considered with its subspace topology.


isotope

en-verb

isotope (third-person singular simple present isotopes, present participle isotoping, simple past and past participle isotoped)

isotopes (present singular third-person)
isotoping (participle present)
isotoped (participle past)
isotoped (past)

(topology, transitive) To define or demonstrate an isotopy of (one map with another).


compact

en-adj

compact (comparative more compact or compacter, superlative most compact or compactest)

more compact (comparative)
compacter (comparative)
most compact (superlative)
compactest (superlative)

From Middle French compact, from Latin compāctus, perfect passive participle of compingō (“join together”), from com- (“together”) + pangō (“fasten”), from Proto-Indo-European *pag- (“to fasten”).

massive compact halo object

massive and compact halo object, massive astronomical compact halo object, massive astrophysical compact halo object

sequentially compact
sequentially compact

(topology, not comparable) Of a topological space:

Such that every open cover has a finite subcover. In a metric space, this is equivalent to being sequentially compact. In metric spaces with the Heine-Borel property, this is equivalent to being closed and bounded.

Compact in the above sense and moreover Hausdorff.


surgery

en-noun

surgery (countable and uncountable, plural surgeries)

/ˈsɜːd͡ʒəɹi/

From Middle English surgerie, from Old French surgerie, from Latin chirurgia, from Ancient Greek χειρουργία (kheirourgía), from χείρ (kheír, “hand”) + ἔργον (érgon, “work”). Doublet of chirurgy.

(topology) The production of a manifold by removing parts of one manifold and replacing them with corresponding parts of others.


descent

descent (countable and uncountable, plural descents)

From Middle English and Anglo-Norman descente, from Anglo-Norman descendre (“to descend”); see descend. Compare ascent, ascend. Doublet of desant.

(topology) A particular extension of the idea of gluing.


covering

covering (countable and uncountable, plural coverings)

From Middle English coverynge, -inge, keverynge, -inge [verbal noun of coveren, keveren (“to cover”)]; or, verbal noun from cover (verb) + -ing.

(topology, of a given open set U) A collection of sets U_i such that their union contains U; such a collection with the additional property that every U_i is open; such a collection with the additional property that every U_i is contained in U.

(topology) A special kind of map that establishes a relationship between two topological spaces in the sense that, under the action of the map, the one looks locally like several copies of the second: Formally, a continuous map π:E→X between topological spaces such that there exists, for every point x in X, a discrete space D_x and an open neighborhood U of x such that π⁻¹(U)= displaystyle ⨆_(d∈D_x)V_d and π|_(V_d):V_d→U is a homeomorphism for every d∈D_x. See Covering space on Wikipedia.Wikipedia


Möbius strip

August Ferdinand Möbius
Johann Benedict Listing
Möbius strip

Named after August Ferdinand Möbius (1790–1868), one of two German mathematicians—the other being Johann Benedict Listing (1808–1882)—who independently discovered the strip at the same time (1858).

Like a mind moving on a Möbius strip of events, one's consciousness goes over the same territory again and again without ever becoming aware of the nature of awareness.

1981, William Irwin Thompson, The Time Falling Bodies Take to Light: Mythology, Sexuality and the Origins of Culture, page 8:

1991, Theoni Pappas, More Joy of Mathematics, Wide World Publishing, page 117,
B. F. Goodrich Co. has a patent for a Möbius strip conveyor belt. It lasts longer since the wear and tear is spread uniformly over the entire surface.

A geometrical model of electron (positron) structure is presented based upon an object of two linked loops obtained by cutting a Möbius strip asymmetrically.

1992, Speculations in Science and Technology, Volume 15, Elsevier Sequoia, page 221:

Proust's novel grew and deepened into over 3,000 pages of everything and nothing: a Möbius strip of profundity twisting into mundanity, mundanity twisting into profundity.

2013 November 16, Patrick McGuinness, “Who's afraid of Marcel Proust?”, in The Daily Telegraph (Review), page R20:

(topology) A one-sided surface formed by identifying two opposite edges of a square in opposite senses;

A one-sided surface formed by identifying two opposite edges of a square in opposite senses;
a narrow strip given a half twist and joined at the ends, forming a three-dimensional embedding of said surface.


orientation

orientation (countable and uncountable, plural orientations)

/ˌɔɹiɛnˈteɪʃən/

From French orientation. By surface analysis, orient + -ation.

(analytic geometry, topology, countable) The designation of a parametrised curve as "positively" or "negatively" oriented; the analogous description of a surface or hypersurface.


virtually

en-adv

virtually (comparative more virtually, superlative most virtually)

more virtually (comparative)
most virtually (superlative)

[ˈvɝt͡ʃuəɫi]

From Middle English vertually; equivalent to virtual + -ly.

Another class of small manifolds consists of those that are not virtually Haken (a manifold is virtually Haken if it is finitely covered by a Haken manifold).

2002, Steven Boyer, “Dehn surgery on knots” (chapter 4), in R.J. Daverman, R.B. Sher, editors, Handbook of Geometric Topology, page 206:

We call a hyperplane Y in a VH-complex X virtually clean if there exists a finite-sheeted covering map p : X̂ → X and a connected component Y ⊆ p⁻¹Y such that the hyperplane Ŷ of X̂ is clean.

2017, Martin R. Bridson, Henry Wilton, “Hyperbolic Geometry and Geometric Group Theory”, in Advanced Studies in Pure Mathematics, volume 73, page 40:

(topology) Of a covering space of finite degree.


loop

From Middle English loupe (“noose, loop”), earlier lowp-knot (“loop-knot”), of North Germanic origin, from Old Norse hlaup (“a run”), used in the sense of a "running knot", from hlaupa (“to leap”), ultimately from Proto-Germanic *hlaupaną (“to leap, run”). Compare Swedish löp-knut (“loop-knot”), Danish løb-knude (“a running knot”), Danish løb (“a course”). More at leap. The verb is derived from the noun.

der

(topology) A path that starts and ends at the same point.

silmukka

topology: path that starts and ends at the same point


subcover

The open intervals cover the real numbers; the open intervals of the form (x, x+1) are a subcover.

(topology) A cover which is a subset of another cover.


companion

From Middle English companion, from Old French compaignon (“companion”) (modern French compagnon), from Late Latin compāniōn- (nominative singular compāniō, whence French copain), from com- + pānis (literally, with + bread), a word first attested in the Frankish Lex Salica as a calque of a Germanic word, probably Frankish *galaibo, *gahlaibō (“messmate”, literally “with-bread”), from Proto-Germanic *gahlaibô. Compare also Old High German galeipo (“messmate”) and Gothic 𐌲𐌰𐌷𐌻𐌰𐌹𐌱𐌰 (gahlaiba, “messmate”); and, for the semantics, compare Old Armenian ընկեր (ənker, “friend”, literally “messmate”). More at co-, loaf. Displaced native Old English ġefēra (literally “fellow traveler”). Doublet of company. Compare mate.

noncog

(topology) A knot in whose neighborhood another, specified knot meets every meridian disk.


PL

piecewise linear
piecewise linear#English

(topology) Initialism of piecewise linear.


annulus

Learned borrowing from Medieval Latin ānnulus, a misspelling of Latin ānulus (“ring, especially one worn on a finger”), from ānus (“ring”) (from Proto-Indo-European *h₁eh₂no- (“ring”)) + -ulus (diminutive suffix).
The plural form annuli is a learned borrowing from Medieval Latin ānnulī.

lbor

(topology) Any topological space homeomorphic to the region in a plane between two concentric circles of different radii.


embedding

(mathematics) A map which, in any of several technical senses, represents the containment of one structure inside another.

(topology) A continuous map which is also a homeomorphism between its domain and its image (considered with the subspace topology induced by its codomain).


topology

en-noun

topology (countable and uncountable, plural topologies)

From Late Latin topologia, from Ancient Greek τόπος (tópos, “place, locality”) + -(o)logy (“study of, a branch of knowledge”).

der

A set X equipped with a topology #x5C;tau is called a topological space and denoted (X,#x5C;tau).

The subsets of a set X which constitute a topology are called the open sets of X.

It is easy to verify that the topology generated by these basic open sets coincides with the definition of the topology on #x5C;widehat#x7B;V#x7D; above, for the Zariski topology and the sheaf of functions #x5C;operatorname#x7B;val#x7D;(f),f regular.

2016, Ehud Hrushovski, François Loeser, Non-Archimedean Tame Topology and Stably Dominated Types (AM-192), Princeton University Press, page 43:

(topology) Any collection τ of subsets of a given set X that contains both the empty set and X, and which is closed under finitary intersections and arbitrary unions.


trefoil

From Middle English trefoil, from Old French trifoil, trefeul, from Latin trifolium, from tri- (“three”) + folium (“leaf”).


pointed

en-adj

pointed (comparative more pointed, superlative most pointed)

more pointed (comparative)
most pointed (superlative)

The class of all pointed spaces forms a category - Top_• - with basepoint preserving continuous maps as morphisms.

(topology, algebraic topology, of a topological space) That has a named, but otherwise arbitrary, point (called the basepoint) that remains unchanged during subsequent discussion and is kept track of during all operations.


connected

connected (comparative more connected, superlative most connected)

more connected (comparative)
most connected (superlative)

As a combinatorial subject, the connected dominating set has been studied in as early as 1970s.

2012, Ding-Zhu Du, Peng-Jun Wan, Connected Dominating Set: Theory and Applications, →ISBN, page v:

(mathematics, topology, of a topological space) That cannot be partitioned into two nonempty open sets.

souvislý

topology: cannot be partitioned into two nonempty open sets


Klein bottle

Calque of German Kleinsche Flasche, named after German mathematician Felix Klein (1849–1925), who first described such a surface in 1882.

named-after

At this level Atman and Brahman are like a Klein bottle or a Möbius strip: there is no edge where one can say that this is Atman, but this is Brahman.

1981, William Irwin Thompson, The Time Falling Bodies Take to Light: Mythology, Sexuality and the Origins of Culture, London: Rider/Hutchinson & Co., page 222:

(topology) The closed manifold obtained by identifying the boundary components of the annulus so that the resultant surface is nonorientable.


hyperbolic

hyperbolic partial differential equation

A hyperbolic isometry f has two (distinct) fixed points on ∂ℋ.

2001, A. F. Beardon, "The Geometry of Riemann Surfaces", in, E. Bujalance, A. F. Costa, and E. Martínez, editors, Topics on Riemann Surfaces and Fuchsian Groups, Cambridge, page 6 http://books.google.com/books?id=RjbQdcP7DgwC&pg=PA6&dq=hyperbolic

(geometry, topology, of an automorphism) Whose domain has two (possibly ideal) fixed points joined by a line mapped to itself by translation.

Exactly one hypercycle is a hyperbolic geodesic, and this is called the axis A_f of f.

(topology) Of, pertaining to, or in a hyperbolic space (a space having negative curvature or sectional curvature).


retract

Partly:
* from retract (verb) (see etymology 1); and
* from Late Latin retractus (“a pulling back, retreat; refuge”), from Latin retractus (“withdrawn”), the perfect passive participle of Latin retrahō (“to draw or pull back, withdraw; to bring back; to compel to turn back; to recall; to get back, recover; to hold back, restrain, withhold; to remove, take away; to bring to light again; (Late Latin) to delay”) (see etymology 1) + -tus (suffix forming action nouns from verbs). Doublet of retrait, retreat, and ritratto.


irreducible

en-adj

irreducible (comparative more irreducible, superlative most irreducible)

more irreducible (comparative)
most irreducible (superlative)

/ˌɪɹɪˈdjuːsɪbəl/

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(topology, of a manifold) Not containing a sphere of codimension 1 that is not the boundary of a ball.


lamination

en-noun

lamination (countable and uncountable, plural laminations)

(topology) A foliation of a closed subset of a manifold by subspaces of one dimension less.


compressible

en-adj

compressible (comparative more compressible, superlative most compressible)

more compressible (comparative)
most compressible (superlative)

(topology, of a surface embedded in a 3-dimensional manifold) Containing a circle that does not serve as the boundary of a disk that lies in the surface, but does serve as boundary of a disk that lies in the ambient manifold.


closure

en-noun

closure (countable and uncountable, plural closures)

From Middle English closure, from Old French closure, from Late Latin clausura, from Latin claudere (“to close”); see clausure and cloture (etymological doublets) and close.

7 THEOREM The closure of any set is the union of the set and the set of its accumulation points.

1955 [Van Nostrand Reinhold], John L. Kelley, General Topology, 2017, Dover, page 42, The closure (𝔗-closure) of a subset A of a topological space (X,𝔗) is the intersection of the members of the family of all closed sets containing A. […]

(topology, of a set) The smallest closed set which contains the given set.


Haken

For Wolfgang Haken, who studied these manifolds.

(topology, of a 3-manifold) Irreducible, compact, and containing a non-∂-parallel incompressible surface (besides a sphere or disk).


oriented

en-adj

oriented (comparative more oriented, superlative most oriented)

more oriented (comparative)
most oriented (superlative)

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(geometry, topology) In any of various technical senses, admitting or being equipped with a notion of handedness (i.e. of distinguishing clockwise and anticlockwise). See Orientability on Wikipedia.Wikipedia


pointlessness

en-noun

pointlessness (countable and uncountable, plural pointlessnesses)

(topology, geometry, uncountable) The quality of not mentioning points.


entourage

/ˈɒn.tʊ.ɹɑːʒ/

/ˈɑ̃ː.tʊ.ɹɑːʒ/

/ˈɑn.tə.ɹɑʒ/

Unadapted borrowing from French entourage, from entourer (“to surround”).

ubor

(topology) A binary relation in a uniform space which generalises the notion of two points being no farther apart than a given fixed distance; a uniform neighbourhood.


metric space

(mathematics, mathematical analysis, topology) Any set on which a metric is defined, giving a distance between any two elements.


topological space

en-noun

topological space (plural topological spaces)

In order to obtain "intuitive insight" into special classes of topological spaces we can proceed in several ways, only a few of which will be pursued in this chapter. For instance, we can seek to describe important topological spaces by means of enough of their properties to completely characterize them, up to homeomorphism.

1964, H. J. Kowalsky, translated by Jay E. Strum, Topological Spaces, Academic Press, page 165:

2011, Jonathan A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Springer, Lecture Notes in Mathematics 2032, page xi,
Most of the spaces studied in Algebraic Topology, such as CW-complexes or manifolds, are Hausdorff. In contrast, finite topological spaces are rarely Hausdorff. A topological space with finitely many points, each of which is closed, must be discrete.

If (X,#x5C;tau) is a topological space, then a cover #x5C;mathcalB is open if each B#x5C;in#x5C;mathcalB is an open set. A topological space (X,#x5C;tau) is compact if every open cover of X has a finite subcover.

2013, D. J. H. Garling, A Course in Mathematical Analysis: Volume 2, Cambridge University Press, page 431:

(topology, formally) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);

An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);
(loosely) the set X.


segment

From Latin segmentum (“a piece cut off, a strip, segment of the earth, a strip of tinsel”), from secāre (“to cut”).

der

(topology) Any of the pieces that constitute an order tree.


shelling

en-noun

shelling (countable and uncountable, plural shellings)

(topology) An ordering of the facets of a boundary complex such that the intersection of each facet (other than the first) with the union of all preceding facets is homeomorphic to a ball or sphere. See Shelling (topology)


train track

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(topology) A set of curves lying in a surface, meeting one another at their respective endpoints in a specific way that resembles railroad tracks.


homology

homology (countable and uncountable, plural homologies)

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Henri Poincaré
Homology
MathWorld
singular homology

From homo- + -logy.
In topology, first used by French polymath Henri Poincaré, in the sense (close to what is now called a bordism) of a relation between manifolds mapped into a reference manifold: that is, the property of such manifolds that they form the boundary of a higher-dimensional manifold inside the reference manifold. Poincaré's version was eventually replaced by the more general singular homology, which is what mathematicians now mean by homology.

homologia

biology, psychology: relationship of having a common evolutionary or developmental origin

One encounters a similar situation in homology theory. Beside singular homology, which is a homotopy invariant, and Čech homology, which is a shape invariant, there exists strong homology, which is a strong shape invariant. In the special case of metric compacta, this homology was introduced by N.E. Steenrod in 1940 and is often referred to as the Steenrod homology.

2000, Sibe Mardešić, Strong Shape and Homology, Springer, page v:

Brieskorn homology spheres are a special case of Siefert fibered homology spheres.

2002, Nikolai Saveliev, Invariants of Homology 3-Spheres, Springer, page 2:

The relationship of being homologous; a homologous relationship.

(topology, algebraic topology) A general way of associating a sequence of algebraic objects, such as abelian groups or modules, to a sequence of topological spaces; also used attributively: see Usage notes below.

同调

algebraic topology: way of associating a sequence of groups to a sequence of topological spaces


metrizable

(topology) Of a topological space: for which a metric exists that will induce the original topology.


continuous function

continuous function (plural continuous functions)

(mathematical analysis, topology) a function from one topological space to another, such that the inverse image of any open set is open


morphism

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1982, Israel Program for Scientific Translations (translator), Lev J. Leifman (editor of translation), N. N. Čencov, Statistical Decision Rules and Optimal Inference, American Mathematical Society, Translations of Mathematical Monographs, Volume 53, page 50,
1° The composition of two morphisms is defined if and only if the final object of the first morphism is the initial object of the second. This composition is also a morphism, whose initial object is the initial object of the first morphism and whose final object is the final object of the second.

1992, Terrance Brown (translator), Gil Henriques, Chapter 13: Morphisms and Transformations in the Construction of Invariants, Terrance Brown (translator), Jean Piaget, Gil Henriques, Edgar Ascher (editors), Morphisms and Categories: Comparing and Transforming, page 198,
In certain extreme cases in mathematics, the synthesis of morphisms and of transformations is so intimate that one can speak of a veritable fusion. […] Essentially, categories are sets of morphisms organized into operatory systems.

The proof of toroidalization of morphisms of 3-folds to surfaces in [C3] breaks up into two parts: a reduction to prepared morphisms and then a proof of toroidalization of prepared morphisms from n-folds to [surfaces] in [CK].

2007 November, Steven Dale Cutkosky, “Toroidalization of Dominant Morphisms of 3-Folds”, in Memoirs of the, volume 190, number 890, American Mathematical Society, page 3:

(mathematics, category theory) (formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises.

Briefly, the yellow morphic males can change their status from paired to satellite and from satellite to the paired one. However, they cannot cross into the status of the red morph.[…] The colour morphism of males is proved to be irreversible after its expression at an early stage of ontogeny.

2010, T.J. Pandian, Sexuality in Fishes, page 51:

(biology) Being or having distinct variants of a plant or animal species in the same locale; polymorphism.


direct product

(topology) A topological space analogously formed from two or more (up to an infinite number of) topological spaces.

suora tulo

product of sets — see also Cartesian product


homeomorphism

(topology) a continuous bijection from one topological space to another, with continuous inverse.


convergent

en-adj

convergent (comparative more convergent, superlative most convergent)

more convergent (comparative)
most convergent (superlative)

(mathematical analysis, topology) Of a sequence or series in a topological space, converging to a (finite, proper) limit.


Lie group

Named for Norwegian mathematician Sophus Lie.

1994, Silvio Levy (translator), Albert S. Schwarz, Topology for Physicists, [1989, A. S. Shvarts, Kvantovaya teoriya polya i topologiya], Springer, 1996, 2nd Printing, page 233,
Every connected Lie group is homotopically equivalent to its maximal compact subgroup. This reduces the study of the homotopy and homology of Lie groups to the compact case.

A Lie group is a manifold on which the group manipulations, product and inverse, arc defined. Lie groups play an extremely important role in the theory of fibre bundles and also find vast applications in physics.

2009, Mikio Nakahara, Geometry, Topology and Physics, 2nd edition, Taylor & Francis, page 207:

As is well known, in finite dimensions each Lie group is, at least locally near the identity, completely described by its Lie algebra.

2009, Boris Khesin, Robert Wendt, The Geometry of Infinite-Dimensional Groups, Springer, page 1:

(topology) Any group that is a smooth manifold and whose group operations are differentiable.

Liegruppe

analytic group that is also a smooth manifold


reducible

reducible (comparative more reducible, superlative most reducible)

more reducible (comparative)
most reducible (superlative)

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(topology, of a manifold) Containing a sphere of codimension 1 that is not the boundary of a ball.


Seifert fibre space

en-noun

Seifert fibre space (plural Seifert fibre spaces)

Seifert fiber space
Seifert fiber space#English

(topology, British) Alternative spelling of Seifert fiber space


Seifert fibered space

Seifert fibered space (plural Seifert fibered spaces)

(topology, US) Alternative spelling of Seifert fiber space


second-countable

(topology) Such that its topology has a countable base, said of a topological space.


fibre bundle

(topology, British, Canada) Alternative spelling of fiber bundle


Seifert fibred space

Seifert fibred space (plural Seifert fibred spaces)


Seifert fiber space

Seifert fiber space (plural Seifert fiber spaces)

(topology, chiefly US) A fiber bundle whose fiber is the circle and whose base space is a two-dimensional orbifold.


homology sphere

homology sphere (plural homology spheres)

(topology) A manifold whose homology is the same as that of some sphere of the same dimension


Betti number

A calque of French nombre de Betti, coined in 1892 by Henri Poincaré; named after Italian mathematician Enrico Betti in recognition of an 1871 paper.

Poincaré proved that Betti numbers are invariants and used them to extend Euler's polyhedral formula to higher dimensional spaces.

1979 [W. H. Freeman & Company], Michael Henle, A Combinatorial Introduction to Topology, 1994, Dover, page 163,
Prove that, for compact surfaces, the zeroth Betti number is the number of components of the surface, where a component is a connected subset of the surface, such that any larger containing subset is not connected.

2007, Oscar García-Prada, Peter Beier Gothen, Vicente Muñoz, Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles, American Mathematical Society, page 7,
PROPOSITION 2.1. Fix the rank r. For different choices of degrees and generic weights, the moduli spaces of parabolic Higgs bundles have the same Betti numbers.

The goal is to compute Betti numbers in 2D and 3D image partitions using the practical definition of Betti numbers. Thus, depending on the dimension of the topological map, we count the number of connected components, the number of tunnels, and the number of cavities to obtain the Betti numbers.[…]The number of connected components of region r in a 2D image partition is equal to the first Betti number b#x5F;0(r).

2012, Guillaume Damiand, Alexandre Dupas, “12: Combinatorial Maps for Image Segmentation”, in Valentin E. Brimkov, Reneta P. Barneva, editors, Digital Geometry Algorithms, Springer, page 380:

(topology, algebraic topology) Any of a sequence of numbers, denoted bₙ, which characterise a given topological space K by giving, for each dimension, the number of holes in K of said dimension; (formally) the rank of the nth homology group, Hₙ, of K.

Betti number
Enrico Betti
Henri Poincaré

Bettital

number that characterises a topological space with respect to a given dimension


homoeomorphism

(British spelling, topology) Alternative spelling of homeomorphism


homoeomorphic

(British, topology) Alternative spelling of homeomorphic


T₂

(topology, of a topological space) Such that any two points have disjoint neighborhoods.


Hausdorff

Named after German mathematician Felix Hausdorff (1868–1942).

named-after

(of a topological space) Such that any two distinct points have disjoint neighborhoods.


topological group

en-noun

topological group (plural topological groups)

(topology) A group which is also a topological space and whose group operations are continuous functions.


fiber bundle

Coined as fibre bundle by American mathematician Norman Steenrod in 1951, The Topology of Fibre Bundles. The related usages fiber and fiber space probably derive (as calques respectively of German Faser and gefaserter Räume) from 1933, Herbert Seifert, “Topologie dreidimensionaler gefaserter Räume,” Acta Mathematica, 60, (1933), 147-238.

A Möbius strip is a fiber bundle which looks locally (i.e., over a connected proper subset of its base space) like the corresponding part of a cylinder S¹#x5C;times#x5B;0,1#x5D; (a Möbius strip and a cylinder have isomorphic base spaces). A Klein bottle is a fiber bundle which looks locally like the corresponding part of a torus S¹#x5C;timesS¹ (again they could be thought of as sharing the same base space S¹; cutting out even a single point of that base space makes the cut Klein bottle isomorphic to the cut torus).

In general, a fiber bundle consists of a set of mutually disjoint fibers “over” a base space, which indexes the fibers; there is a copy of some fiber on top of, or projecting (“canonically”) onto each point of the base space.

In the 1960s, some physicists including E. Lubkin and A. Trautman recognized that interaction potentials can be represented by connections on principal fiber bundles. In 1975, T. T. Wu and C. N. Yang used the fiber bundle method to solve a problem on magnetic monopoles.

1995, Sunny Y. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, page 214:

One proposed general model for high-level scientific data uses fiber bundles, which Butler and Pendley [22] proposed in 1989.[…] A fiber bundle is the Cartesian product of the fibers and the base space; in other words, it is the collection of valid data ranges for the base space.

2001, John M. May, Parallel I/O for High Performance Computing, Morgan Kaufmann Publishers, page 236:

Finding the right notion of fiber bundle for diffeology [Igl85] has been a question raised by the study of the irrational torus T#x5F;#x5C;alpha [Dolg85].

2013, Patrick Iglesias-Zemmour, Diffeology, American Mathematical Society, page 229:

(American spelling, topology, category theory) An abstract object in topology where copies of one object are "attached" to every point of another, as hairs or fibers are attached to a hairbrush. Formally, a topological space E (called the total space), together with a topological space B (called the base space), a topological space F (called the fiber), and surjective map π from E to B (called the projection or submersion), such that every point of B has a neighborhood U with π⁻¹(U) homeomorphic to the product space U × F (that is, E looks locally the same as the product space B × F, although its global structure may be quite different).

Herbert Seifert
Norman Steenrod
Project Euclid
fiber bundle


homotopy

homotopy (countable and uncountable, plural homotopies)

From Ancient Greek ὁμός (homós, “same, similar”) + τόπος (tópos, “place”); earliest known use in print in 1922, Oswald Veblen, Analysis Situs.

homotopie

(topology) theory associating a system of groups with each topological space

The concept of homotopy represents a formalisation of the intuitive idea of a smooth deformation of one curve into another.

An integer M is called an exponent for the torsion of an abelian group G if M * (torsion of G) = 0. We say that M is a homotopy exponent for a space X if M is an exponent for πₖ (X) for all k.

1998, Paul Sellick, “Space Exponents for Loop Spaces of Spheres”, in William G. Dwyer, editor, Stable and Unstable Homotopy, American Mathematical Society, page 279:

A graded Lie algebra arises from these maps via the Samelson product in homotopy, the so-called homotopy Lie algebra which is discussed below.

2001, F. R. Cohen, S. Gitler, “Loop-spaces of configuration spaces, braid-like groups, and knots”, in Jaume Aguadé, Carles Broto, Carles Casacuberta, editors, Cohomological Methods in Homotopy Theory, Springer (Birkhäuser), page 63:

2010, Vladimir G. Turaev, Homotopy Quantum Field Theory, European Mathematical Society, page xi,
In this monograph we apply the idea of a TQFT to maps from manifolds to topological spaces. This leads us to a notion of a (d + 1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed oriented d-dimensional manifolds and compact oriented (d + 1)-dimensional cobordisms endowed with maps to a given space X.

(topology) A continuous deformation of one continuous function or map to another.

homotopi

(topology) continuous deformation of one continuous function to another

(topology) A theory associating a system of groups with each topological space.

(topology) A system of groups associated with a topological space.

homotopie

(topology) system of groups associated with a topological space


top-dimensional

(topology) Having the same dimension as the ambient topological space.


cube with handles

en-noun

cube with handles (plural cubes with handles)

(topology) A three-dimensional handlebody


handlebody

(topology) A top-dimensional submanifold of Euclidean space comprising a ball with handles attached to it along its boundary.


submanifold

(topology) A manifold which is a subset of another, so that the inclusion function is an embedding or, sometimes, an immersion.


piecewise linear

(topology) Having a defining atlas such that there is a piecewise linear homeomorphism between any pair of intersecting atlas charts: said of manifolds


pretzel link

(topology) A link with a standard projection comprising several tangles, each a braid on two strands, with the tangles arranged in some cyclic order.


pretzel knot

(topology) A pretzel link which is a knot.


gluing

gluing (countable and uncountable, plural gluings)

(topology) A connection between the edges of two planes, or between the points at the ends of two lines.

склейка

a connection between the edges of two planes, or between the points at the ends of two lines


Dehn surgery

Dehn surgery (countable and uncountable, plural Dehn surgeries)

(topology) The removal of a solid torus from a 3-dimensional manifold, followed by its being replaced via a homeomorphism of its boundary.

(topology, specifically) The same, but with the manifold in question being the 3-dimensional sphere.


simple connectedness

(topology) The state of being simply connected.


simply connected

(topology, of a topological space) Having its fundamental group a singleton.

单连通的

in topology, having its fundamental group a singleton


solid torus

(topology) The topological space that is a Cartesian product of the two-dimensional disk and the circle.


(topology, of a topological space) Having a singleton as its fundamental group.


fundamental group

fundamental group (plural fundamental groups)

From the definition it will be clear the group is a topological invariant of X; i.e., if two spaces are homeomorphic, their fundamental groups are isomorphic.

1991, William S. Massey, A Basic Course in Algebraic Topology, Springer, page 35:

Therein came into focus the far-reaching vision that the perspective of non-abelian fundamental groups could lead to a fundamentally new understanding of deep arithmetic phenomena, including the arithmetic theory of moduli and Diophantine finiteness on hyperbolic curves.

2011, John Coates, Minhyong Kim, Florian Pop, Mohamed Saïdi, Peter Schneider, editors, Non-abelian Fundamental Groups and Iwasawa Theory, Cambridge University Press, page vii:

During the more than 100 years of its existence, the notion of the fundamental group has undergone a considerable evolution.

2012, Jakob Stix, editor, The Arithmetic of Fundamental Groups: PIA 2010, Springer, page ix:

(topology) For a specified topological space, the group whose elements are homotopy classes of loops (images of some arbitrary closed interval whose endpoints are both mapped to a designated point) and whose group operation is concatenation.

grup fonamental

group of equivalence classes of the homotopies of loops in a given topological space


Euler characteristic

Euler characteristic (plural Euler characteristics)

A polygon's or polyhedron's Euler characteristic is just the number of corners minus the number of edges plus the number of faces.

(topology, of a topological space) The sum of even-dimensional Betti numbers minus the sum of odd-dimensional ones.


flag complex

(topology) A simplicial complex where every collection of pairwise adjacent vertices spans a simplex.


simplicial complex

simplicial complex (plural simplicial complexes)

(topology) A collection of simplices of various dimensions, connected to one another according to certain rules.


discrete topology

discrete topology (plural discrete topologies)

In the discrete topology all sets are open (but they are also all closed).

An infinite set with discrete topology is not compact.

(topology) A topology on a set consisting of all subsets of that set.


nulhomotopic

(topology, of a map) Homotopic to a constant map.


CW complex

The C stands for "closure-finite", and the W for "weak topology".

(topology) A collection of balls of various dimensions, with higher-dimensional ones mapped along their respective boundaries to lower-dimensional ones.


homotopic

f and g are homotopic.

(topology, of two continuous maps) Such that there is a homotopy (a continuous deformation) taking one to the other.

homotopisch

(topology) such that there is a homotopy taking one to the other


nonorientable

The Möbius strip is a nonorientable surface.

Through every point of a nonorientable surface there passes a Möbius-like nonorientable strip (with an odd number of half-twists) which is part of that surface.


pair of pants

A pair of pants . pair of pants (canonical)
pairs of pants (plural)

(topology) The orientable surface of genus zero having three boundary components.


abstract harmonic analysis

abstract harmonic analysis (uncountable)

1970, Edwin Hewitt, Kenneth A. Ross, Abstract Harmonic Analysis: Volume II: Structure and Analysis for Compact Groups; Analysis on Locally Compact Abelian Groups, Springer, 2nd reprint, 2002, page 547,
WIENER [1] goes well beyond analysis of periodic functions on R or even functions in 𝔏₂(R); this important paper has yet to be interpreted and assimilated into the standard repertory of abstract harmonic analysis.

The present book is a text book for a graduate course on abstract harmonic analysis and its applications.

2014, Anton Deitmar, Siegfried Echterhoff, Principles of Harmonic Analysis, 2nd edition, Springer, page v:

2015, Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd edition, Taylor & Francis (CRC Press):

(harmonic analysis, representation theory, topology) A branch of harmonic analysis dealing with generalised Fourier transforms that act on functions defined on Hausdorff locally compact topological groups.

harmonic analysis; representation theory; harmonic analysis; representation theory


subnet

(topology) The abstraction of a sequence.


compactification

compactification (countable and uncountable, plural compactifications)

(topology) Any of various procedures of enlarging a topological space to make it compact.

kompaktifikation

procedure of enlarging a topological space to make it compact

kompaktifikation

topological space enlarged to become compact

(topology) The space resulting from any such procedure.


connected component

connected component (plural connected components)

(topology and graph theory) A connected subset that is, moreover, maximal with respect to being connected.


Reebless

(topology) (of a foliation) Not containing a Reeb component.


Reeb component

Introduced by the French mathematician Georges Reeb (1920–1992).

(topology) A solid torus whose boundary is a leaf of a foliation of the ambient 3-dimensional manifold.


closed set

Coordinate term: open set

(topology) A set whose complement is open.


open set

(topology) Informally, a set such that the target point of a movement by a small amount in any direction from any point in the set is still in the set; exemplified by a full circle without its boundary.

(topology, mathematical analysis, restricted to metric spaces) A set which can be described as an (arbitrary) union of open balls. Equivalently, a set such that for every point in it, there is an open ball centered at that point, such that that open ball is contained by the set.

(topology) Most generally, a member of the topology of a given topological space.


weakly contractible

(of a topological space) Having all its homotopy groups trivial.


leafspace

(mathematics) A topological space that is the quotient of a laminated or foliated manifold by identification of each leaf, and each closed complementary region, to a point.


covering space

Let (X, p) be a covering space of X such that X is simply connected.

2019, William S. Massey, A Basic Course in Algebraic Topology, page 132:

(topology) The domain of a covering of a topological space; the covering together with its domain. (See Usage notes at covering).


homotope

en-verb

homotope (third-person singular simple present homotopes, present participle homotoping, simple past and past participle homotoped)

homotopes (present singular third-person)
homotoping (participle present)
homotoped (participle past)
homotoped (past)

(topology, transitive) To define or demonstrate a homotopy of (one map with another).

(topology, transitive) More loosely, to exhibit a homotopy equivalence between two spaces.


missing boundary

(of a compact topological space with boundary) Having had some but not all of its boundary removed.


alternating knot

en-noun

alternating knot (plural alternating knots)

(topology) A knot that has a knot diagram in which, in traveling along the knot, one passes through intersections alternatingly: over, under, over, under, etc.


knot diagram

(topology) A representation in the plane of a knot, with no triple points.


fibration

(algebraic topology) A continuous mapping satisfying the homotopy lifting property with respect to any space.


crisping

For two vague regions, one topological relation exists between any pair of crispings. There are a number of relations possible between the members of the sets of crisp regions on which the vague regions stand.

2003, Wenzhong Shi, Peter Fisher, Michael F. Goodchild, Spatial Data Quality:

(topology) A crisp, clearly-defined object that exists within a more vaguely defined object.


quotient space

Thus if #x5C;mathfrak#x7B;D#x7D; is an arbitrary decomposition of a space X into mutually disjoint subsets, then the corresponding quotient space is obtained by 'shrinking' or 'identifying' each member of #x5C;mathfrak#x7B;D#x7D; to a single point. For this reason, the quotient spaces are sometimes called identification spaces and quotient maps as^([sic]) identification maps.

1983, K. D. Joshi, Introduction to General Topology, New Age International, page 129:

1989, unnamed translator, Nicolas Bourbaki, Elements of Mathematics: General Topology: Chapters 1-4, [1971, N. Bourbaki, Éléments de Mathématique: Topologie Générale 1-4, Masson], Springer, page 110,
PROPOSITION 6. Every quotient space of a connected space is connected.

The quotient space obtained from such a relation is called extended Reeb (ER) quotient space. Moreover, the ER quotient space, which is an abstract sub-space of M* and is independent from the geometry, may be represented as a traditional graph, which is called the extended Reeb graph (ERG).

2004, Silvia Biasotti, Bianca Falcidieno, Michela Spagnuolo, “6: Surface Shape Understanding Based on Extended Reeb Graphs”, in Sanjay Rana, editor, Topological Data Structures for Surfaces: An Introduction to Geographical Information Science, Wiley, page 96:

equivalence relation
equivalence relation

(topology and algebra) A space obtained from another by identification of points that are equivalent to one another in some equivalence relation.

identification space

space composed of points equivalent to each other in some relation


ℝ-order tree

(topology) An order tree whose segments are homeomorphic to segments of the real line and that is the union of countably many segments.


order tree

(topology) A set that is the union of subsets (called segments), each of which is totally ordered, such that the segments fit together a certain way.


semiregular

LL-Q1860 (eng)-Flame, not lame-semiregular.wav

A space is semiregular iff the regularly open sets (3D) form a base for the topology.[…]Every space X can be embedded in a semiregular space.

1970, Stephen Willard, General Topology, published 2004, page 98:

(topology, of a topological space) Whose regular open sets form a base.


Hausdorff space

Hausdorff space (plural Hausdorff spaces)

named-after

Named after German mathematician Felix Hausdorff (1868–1942)

More generally, each compact subset of a Hausdorff space is closed.[…]Metric spaces and compact Hausdorff spaces enjoy an even stronger separation property; in either case, disjoint closed sets can always be separated by disjoint open sets.

2005, N. L. Carothers, A Short Course on Banach Space Theory, Cambridge University Press, page 167:

In 1953, Grothendieck [G] characterized locally convex Hausdorff spaces which have the Dunford-Pettis property and used this property to characterize weakly compact operators u : C(K) → F, where K is a compact Hausdorff space and F is a locally compact Hausdorff space (briefly, lcHs) which is complete.

2008, Thiruvaiyaru V. Panchapagesan, The Bartle-Dunford-Schwartz Integral, Springer (Birkhäuser), page ix:

Our construction of βD is a special case of more general constructions in which compact Hausdorff spaces are obtained using sets which are maximal subject to having certain algebraic properties.

2012, Neil Hindman, Dona Strauss, Algebra in the Stone-Cech Compactification: Theory and Applications, 2nd edition, Walter de Gruyter, page 83:

(topology) A topological space in which for any two distinct points x and y, there is a pair of disjoint open sets U and V such that x∈U and y∈V.

T₂ space

topological space in which distinct points are contained in distinct open sets


Polish space

They were first extensively studied by Polish topologists and logicians, such as Sierpiński, Kuratowski, and Tarski.

ℝⁿ is a Polish space

completely metrizable
completely metrizable

(topology) A separable, completely metrizable topological space


limit point

is an example of a function having infinitely many essential singularities with a limit point at z=0. It is an easy matter to give examples of "elementary" functions whose singularities have a countable number of limit points.

1962 [Ginn and Company], Einar Hille, Analytic Function Theory, Volume 2, 2005, American Mathematical Society, page 19, The function sin ( cot 1/z)

If z_0 is a limit point of S, then every ϵ-neighborhood of z_0 must contain infinitely many points of S.

1970 [Macmillan], John W. Dettman, Applied Complex Variables, 1984, Dover, unnumbered page, Let S' be the set of limit points of a set S. Then the closure ◌̅S of S is S∪S'. […]

As the next example shows, the set of zeros may well have a limit point not belonging to the domain.

2006, Ivan Francis Wilde, Lecture Notes on Complex Analysis, Imperial College Press, page 153:

(topology) Given a subset S of a given topological space T, any point p whose every neighborhood contains some point, distinct from p, which belongs to S.

verdichtingspunt

point whose every neighborhood contains a point in a given subset


Mazur game

(topology) A topological game invented by the Polish mathematician Stanisław Mazur in order to illustrate the difference between sets of the first category and second category.


complementary region

en-noun

complementary region (plural complementary regions)

(topology) A connected component of the complement in a manifold of a lamination of that manifold.


accumulation point

accumulation point (plural accumulation points)

LEMMA 5.2 Let X be a Hausdorff space and A a subset of X. A point a#x5C;inX is an accumulation point of A if and only if a is a limit point of A.

1975, Bert Mendelson, Introduction to Topology, 3rd edition, New York: Dover Publications, Inc., published 1990, →ISBN, →OCLC, §5.3, page 173:

2008, Brian S. Thomson, Andrew M. Bruckner, Judith B. Bruckner, Elementary Real Analysis, Volume 1, Thomson-Bruckner (ClassicalRealAnalysis.com), 2nd Edition, page 153,
Definition 4.9 (Closed): The set E is said to be closed provided that every accumulation point of E belongs to the set E.
Thus a set E is not closed if there is some accumulation point of E that does not belong to E. In particular, a set with no accumulation points would have to be closed since there is no point that needs to be checked.

(topology, "of" a subset of a topological space) Given a subset S of a topological space X, a point x whose every neighborhood contains at least one point distinct from x that belongs to S.


deformation retract

deformation retract (plural deformation retracts)

(topology) A subspace such that there is a retraction onto it of the ambient space homotopic to the identity function on the ambient space.


dunce hat

(topology) The topological space obtained by identifying the sides comprising the boundary of a triangle, two sides with consistent orientation and the third with the reverse orientation.


identification space

identification space (plural identification spaces)

(topology) A space obtained from another by identification of points that are equivalent to one another in some equivalence relation.


factor space

(topology, idiomatic) A space obtained from another by identification of points that are equivalent to one another in some equivalence relation.

(topology, literally, relatively rare) A space which is a factor (multiplicand) in a product space.


open book

open book decomposition
open book decomposition

(mathematics, topology) An open book decomposition.


open book decomposition

open book decomposition (plural open book decompositions)

(mathematics, topology) A decomposition of a 3-manifold into a link and a fibration over the circle by surfaces bounded by that link.


Lindelöf space

Named for the Finnish mathematician Ernst Leonard Lindelöf.

(topology) A topological space such that any open cover of the space has a countable subcover.


cinquefoil

From Old French cinqfoil, from cinq (“five”) + foil (“leaf”).

(topology) A particular knot of five crossings.


chain complex

boundary homomorphism
boundary homomorphism

(algebraic topology) A sequence of Abelian groups (or modules), together with a sequence of homomorphisms (called boundary homomorphisms) between consecutive groups, such that the image of one homomorphism is contained in the kernel of the next.


δ-hyperbolic

(of a topological space) In which every geodesic triangle satisfies that every point on the triangle is within a distance of δ (some constant) of each side of the triangle.


interior point

(mathematics, topology) A point in a set S that has a neighbourhood which is contained in S.


supercompact

en-adj

supercompact (comparative more supercompact, superlative most supercompact)

more supercompact (comparative)
most supercompact (superlative)

LL-Q1860 (eng)-Flame, not lame-supercompact.wav

(mathematics, topology, of a topological space) Having a subbasis such that every open cover of the topological space from elements of the subbasis has a subcover with at most two subbasis elements.


open ball

(topology, mathematical analysis, restricted to metric spaces) The set of all points in a metric space whose distance from a given point ("the open ball's center") is less than a given length ("the open ball's radius").


shellable

(topology) For which a shelling is possible.


open cover

(topology) A cover whose members are all open sets.


regular space

(topology) A Hausdorff space with the additional property that for every closed set of that space and every point disjoint from that set, there are a disjoint pair of open sets which contain the closed set and the point, respectively.


locally compact

(topology) Of a topological space: such that, for every point of the space, there is a neighborhood of that point whose closure is compact.


normal space

(topology) A regular space with the additional property that for every disjoint pair of closed sets in that space, there is a disjoint pair of open sets which contain the closed sets, respectively.


cutpoint

Every point of a line is a cutpoint, but no point of a circle is a cutpoint.

(topology) A point of a connected space whose removal causes the resulting space to be disconnected.


trefoil knot

From trefoil + knot, from being a knot theory knot, where its simplest symmetrical configuration appears like a trefoil.
File:Trefoil-non-3-symm.svg|A trefoil knot lacking three-fold symmetry
File:Tricoloring.png|The trefoil knot with each lobe highlighted
File:Trefoil Knot.gif|Trefoil knot

(topology) A knot formed by joining the two loose ends of a overhand knot to form a knotted loop; the simplest non-trivial knot.


Kolmogorov space

en-noun

Kolmogorov space (plural Kolmogorov spaces)

Named in honor of Andrey Kolmogorov (1903–1987), a Soviet mathematician.

The Kolmogorov quotient of any topological space is a Kolmogorov space.

(topology) A topological space for which any pair of distinct points are also topologically distinct, i.e., there is an open set which contains one of the points but not the other one.


Kolmogorov

Named after Russian mathematician Андре́й Никола́евич Колмого́ров (Andréj Nikolájevič Kolmogórov) (1903–1987)

lang

Андре́й Никола́евич Колмого́ров

Андре́й Никола́евич Колмого́ров

xlit

Андре́й Никола́евич Колмого́ров

Андре́й Никола́евич Колмого́ров

named-after

Андре́й Никола́евич Колмого́ров

(of a topological space) Such that any two distinct points are topologically distinguishable, i.e., there is an open set containing one of the points which does not contain the other point.


cap product

A reference to the symbol used, ∩, in this context called "cap"; compare cup product.

(topology) A method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q.


Poincaré space

Any closed, orientable, connected manifold is a Poincaré space.

(topology) An n-dimensional topological space with a distinguished element µ of its nth homology group such that taking the cap product with an element of the kth cohomology group yields an isomorphism to the (n − k)th cohomology group.


discrete set

(topology) A set of points of a topological space such that each point in the set is an isolated point, i.e. a point that has a neighborhood that contains no other points of the set.


cactoid

It is well known that a Peano space is a cactoid if and only if each of its true cyclic elements is a 2-sphere.

1951, John William Theodore Youngs, The Representation Problem for Fréchet Surfaces, page 27:

(topology) A locally connected continuum that is the closure of the union of an at most countable number of spheres and simple arcs, of which every simple closed contour is contained in one and only one of the spheres.


Smale's paradox

(topology) The non-intuitive fact that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease.


subspace topology

en-noun

subspace topology (plural subspace topologies)

(topology) The topology of a subset S of a topological space X which is obtained by considering any subset of S to be an open set if it corresponds to the intersection of S with some open set of X.


relative topology

relative topology (plural relative topologies)

So called because open sets of such a (subspace) topology are defined in terms of open sets in the topology of the enclosing space.


standard topology

(topology) The topology of the real number system generated by a basis which consists of all open balls (in the real number system), which are defined in terms of the one-dimensional Euclidean metric.

(topology) The topology of a Euclidean space ℝⁿ such that any subset of that space is open (i.e. belonging to the topology) if it can be written as a union of open balls from that space.


product topology

product topology (plural product topologies)

(topology) The topology of the Cartesian product of two or more topological spaces which is generated from a basis whose elements are Cartesian products of open subsets of the original spaces, such that for each such Cartesian product only a finite quantity of such subsets are proper.


arcwise-connected

For quotations using this term, see Citations:arcwise-connected.


preregular

(topology, of a space) In which any two topologically distinguishable points can be separated by neighbourhoods


K-theory

From circa 1960. The K stands for German Klasse (“class”).
The theory developed out of algebraic geometry after the 1957 publication of work by German-born French mathematician Alexander Grothendieck.

algebraic K-theory

study of rings generated by vector bundles

K'''-theory as an independent discipline is a fairly new subject, only about 35 years old.

1994, Jonathan Rosenberg, Algebraic K-Theory and Its Applications, Springer, page 1:

K'''-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C^*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K'''-theory (for C^*-algebras) is a pair of functors, called K₀ and K₁, that to each C^*-algebra A associate two Abelian groups K₀(A) and K₁(A).

2000, M. Rørdam, F. Larsen, Flemming Larsen, N. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, page ix:

topological K-theory
topological K-theory

(uncountable, algebra, algebraic geometry, algebraic topology) The study of rings R generated by the set of vector bundles over some topological space or scheme;

The study of rings R generated by the set of vector bundles over some topological space or scheme;
(dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.


Cauchy space

(topology) A space for which notions of a Cauchy sequence and completeness exist.


uniform space

uniform continuity
uniform continuity#English

uniform convergence
uniform convergence#English

(topology) A space for which properties like uniform continuity and uniform convergence can be formulated.

一様空間

a space for which uniform continuity can be formulated


wedge sum

(topology) A one-point union of a family of topological spaces.

yhden pisteen yhdiste

a one-point union of a family of topological spaces


disjoint union

(topology) The disjoint union of the underlying sets of a given family of topological spaces, equipped with a topology.

coproduct

union of topological spaces equipped with a topology


continuum theory

(topology) The branch of topology devoted to the study of continua, a type of metric space.


Riemann sphere

Named after German mathematician Bernhard Riemann.

We use #x5C;hat#x7B;#x5C;Complex#x7D;#x5C;times (resp. #x5C;Complex) to denote the subset #x5C;#x7B;z#x5C;in#x5C;hat#x5C;Complex#x3A;z#x5C;ne 0#x5C;#x7D; (resp. #x5C;#x7B;z#x5C;in#x5C;hat#x5C;Complex#x3A;z#x5C;ne#x5C;infty#x5C;#x7D;) in the Riemann sphere #x5C;hat#x5C;Complex.

2002, Wenhua Zhao, “Some Generalizations of Genus Zero Two-Dimensional Conformal Field Theory”, in Stephen Berman, Paul Fendley, Yi-Zhi Huang, Kailash Misra, Brian Parshall, editors, Recent Developments in Infinite-dimensional Lie Algebras and Conformal Field Theory, American Mathematical Society, page 306:

(topology, complex analysis) The complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.

1967 [Prentice-Hall], Richard A. Silverman, Introductory Complex Analysis, Dover, 1972, page 22,
Every circle γ on the Riemann sphere Σ which does not go through a given point P^*∈Σ divides Σ into two parts, such that one part contains P^* and the other does not.

In order to visualize the point at infinity, we consider the Riemann sphere that has radius 1#x2F;2 and is tangent to the complex plane at the origin (see Figure 1.8). We call the point of contact the south pole (denoted by S) and the point diametrically opposite S the north pole (denoted by N). Let z be an arbitrary point in the complex plane, represented by the point P. We draw the line PN which intersects the Riemann sphere at the unique point P', distinct from N. Conversely, to each point P' on the sphere, other than the north pole N, we draw the line P'N which cuts uniquely one point P in the complex plane. Clearly, there exists a one-to-one correspondence between points on the Riemann sphere, except N, and all the finite points on the complex plane. We assign the north pole N as the point at infinity. With such an assignment, we then establish a one-to-one correspondence between all the points on the Riemann sphere and all the points in the extended complex plane. This correspondence is known as the stereographic projection.

2002, Yue Kuen Kwok, Applied Complex Variables for Scientists and Engineers, Cambridge University Press, page 27:

We find that many mathematicians, even those who specialize in complex analysis and conformal geometry, are not familiar with the inversive distance between pairs of circles in the Riemann sphere.

2003, Philip L. Bowers, Monica K. Hurdal, “Planar Conformal Mappings of Piecewise Flat Surfaces”, in Hans-Christian Hege, Konrad Polthier, editors, Visualization and Mathematics III, Springer, page 6:

extended complex plane
extended complex plane

extended complex numbers
extended complex numbers

(topology, complex analysis) The 2-sphere embedded in Euclidean three-dimensional space and often represented as a unit sphere, regarded as a homeomorphic representation of the extended complex plane and thus the extended complex numbers.

complex projective line

differently constructed set homeomorphic to the Riemann sphere


closed curve

(topology) A map from the circle, S¹, to a topological space.^((McLarty, p. 4))


Stone space

Named after Marshall Harvey Stone (1903–1989), an American mathematician.

For every Boolean algebra there is a Boolean ring corresponding to it. The prime spectrum of a Boolean ring is a Stone space.

topological space
topological space#English

(topology, logic) A topological space constructed from a Boolean algebra, whose points are ultrafilters on the algebra.


Hilbert cube

Named after German mathematician David Hilbert (1862–1943)

named-after

(topology) A metrizable topological space which is the product of a countably infinite number of closed intervals.


Cantor set

Named after German mathematician Georg Cantor (1845–1918).

The Lebesgue null sets include not only all countable sets but many sets having the cardinality of the continuum. We now present the standard example, the Cantor set, which is also of interest for other reasons.

1999, Gerald B. Folland, Real Analysis : Modern Techniques and Their Applications, 2nd edition, New York: John Wiley & Sons, Inc., →ISBN, →OCLC, §1.5, page 38:

mathematical analysis
mathematical analysis

connected component
connected component#English

(mathematical analysis, topology) A subset of an interval formed by recursively removing an interval in the middle of every connected component of the set.


winding number

(geometry, topology, mathematical analysis) The total number of times a given curve travels counterclockwise around a specified point.

indeks

the number of times a curve travels counterclockwise around a point


antipodism

(topology) The relation between points on a sphere which are directly opposite each other; or the function which maps a point to its opposite point.


long line

(mathematics) A topological space, constructed by stringing together an uncountable number of intervals, which is "longer" than the real line.


Brunnian link

Named after Hermann Brunn, whose 1892 article Über Verkettung included examples of such links.

(topology) A nontrivial link that becomes a set of trivial unlinked circles if any one component is removed. In other words, cutting any loop frees all the other loops.


Jordan curve

Named after French mathematician Camille Jordan (1838-1922), who first proved the Jordan curve theorem.

When μ is small and positive, the locus (1) consists of a Jordan curve near each of the Jordan curves belonging to B.

1950, Joseph Leonard Walsh, The Location of Critical Points of Analytic and Harmonic Functions, American Mathematical Society, page 242:

If fis a closed curve that is one-one on [0,1] except for f(0) = f(I), then we say that f is a Jordan curve. A Jordan curve can be seen as a homeomorphism from the circle C onto a subset of the given space.

1992, Konrad Jacobs, Invitation to Mathematics, page 164:

The most general Jordan curves, like the triangle, have the property of dividing the plane into two regions.

2001, Constantin Carathéodory, Theory of Functions of a Complex Variable, volume 1, page 100:

(topology) A non-self-intersecting continuous loop in the plane; a simple closed curve.


distance function

en-noun

distance function (plural distance functions)


noncompactness

Our aim in this paper is to present results of existence of fixed points for continuous operators in Banach spaces using measure of noncompactness under an integral condition.

2015, Vatan Karakaya, Bouzara Nour El Houda, Kadri Dogan, Yunus Atalan, “Existence of fixed points for condensing operators under an integral condition”, in (Please provide the book title or journal name):

(topology) The quality or state of being noncompact.


fibrational

en-adj

fibrational (comparative more fibrational, superlative most fibrational)

more fibrational (comparative)
most fibrational (superlative)

Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory.

2015, Emily Riehl, Dominic Verity, “Kan extensions and the calculus of modules for ∞-categories”, in arXiv:

(algebraic topology) Of or pertaining to a fibration.


hypertorus

LL-Q1860 (eng)-Flame, not lame-hypertorus.wav

(geometry, topology) An N-dimensional equivalent of a torus, where N is a positive integer greater than 3.


precompact

(topology) Forming a relatively compact subspace; Having a closure that is compact.

(topology) Totally bounded; Having a cover that consists of finitely many finite subsets.


supercyclic

LL-Q1860 (eng)-Flame, not lame-supercyclic.wav

(topology) Having a subspace of hypercyclic vectors


quasi-compact

(topology, of a space) In which has every open cover has a finite subcover.


Pachner move

(topology) Any of a set of ways of replacing a triangulation of a piecewise linear manifold by a different triangulation of a homoeomorphic manifold.


pseudoline

LL-Q1860 (eng)-Flame, not lame-pseudoline.wav

(topology) A curve that shares certain topological properties with a line.


angelicity

(topology) The quality of a Hausdorff space of being angelic.


Lindelöf

After the Finnish mathematician Ernst Leonard Lindelöf.

(topology, of a topological space) Such that any open cover has a countable subcover.


multitwist

A Leininger-Reid subgroup cannot be convex cocompact, because it contains a multitwist.

2007, Athanase Papadopoulos, Handbook of Teichmüller Theory - Volume 1, →ISBN, page 407:

(topology) An automorphism that induces a set of Dehn twists around a surface.