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,#92;tau) is a topological space, then a cover #92;mathcalB is open if each B#92;in#92;mathcalB is an open set. A topological space (X,#92;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”).

jaoke

botany: portion of an organ derived from a single cell within the primordium

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


In its original sense "disjunction", probably from the initial letter of Latin vel (“or”).


ℤ

From 𝐙, from abbreviation of German Zahlen (“numbers”). The usage of blackboard bold dates to the 1960s.

(algebra, topology, order theory) The set of integers with some usual structure.


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

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

(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)

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

Henri Poincaré
Homology
MathWorld
singular homology

From Latin homologia, from Ancient Greek ὁμολογία (homología, “agreement, assent”); compare French homologie. By surface analysis, 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

en-noun

continuous function (plural continuous functions)

mathematical analysis
mathematical analysis

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


direct product

suora tulo

product of sets — see also Cartesian product

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


homeomorphism

homeomorphisms (plural)
homoeomorphism (alternative)
homœomorphism (alternative)

(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 (typically infinite) 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)

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

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


baza

bazowy (adjective)
baza (nominative singular)
bazy (nominative plural)
bazy (genitive singular)
baz (genitive plural)
bazie (dative singular)
bazom (dative plural)
bazę (accusative singular)
bazy (accusative plural)
bazą (instrumental singular)
bazami (instrumental plural)
bazie (locative singular)
bazach (locative plural)
bazo (singular vocative)
bazy (plural vocative)

→ Kashubian: baza

→ Silesian: baza

Borrowed from French base. First attested in 1683.

ref

Danuta Lankiewicz (22.06.2015) “BAZA”, in Elektroniczny Słownik Języka Polskiego XVII i XVIII Wieku [Electronic Dictionary of the Polish Language of the XVII and XVIII Century]

(topology) base (set of sets from which a topology is derived)


ympäristö

ympäristö (nominative singular)
ympäristöt (nominative plural)
ympäristö (accusative nominative singular)
ympäristöt (accusative nominative plural)
ympäristön (accusative genitive singular)
ympäristöt (accusative genitive plural)
ympäristön (genitive singular)
ympäristöjen (genitive plural)
ympäristöä (partitive singular)
ympäristöjä (partitive plural)
ympäristössä (inessive singular)
ympäristöissä (inessive plural)
ympäristöstä (elative singular)
ympäristöistä (elative plural)
ympäristöön (illative singular)
ympäristöihin (illative plural)
ympäristöllä (adessive singular)
ympäristöillä (adessive plural)
ympäristöltä (ablative singular)
ympäristöiltä (ablative plural)
ympäristölle (allative singular)
ympäristöille (allative plural)
ympäristönä (essive singular)
ympäristöinä (essive plural)
ympäristöksi (singular translative)
ympäristöiksi (plural translative)
ympäristöttä (abessive singular)
ympäristöittä (abessive plural)
ympäristöin (instructive plural)
See the possessive forms below. (comitative)
ympäristöni (nominative singular)
ympäristöni (nominative plural)
ympäristöni (accusative nominative singular)
ympäristöni (accusative nominative plural)
ympäristöni (accusative genitive singular)
ympäristöni (accusative genitive plural)
ympäristöni (genitive singular)
ympäristöjeni (genitive plural)
ympäristöäni (partitive singular)
ympäristöjäni (partitive plural)
ympäristössäni (inessive singular)
ympäristöissäni (inessive plural)
ympäristöstäni (elative singular)
ympäristöistäni (elative plural)
ympäristööni (illative singular)
ympäristöihini (illative plural)
ympäristölläni (adessive singular)
ympäristöilläni (adessive plural)
ympäristöltäni (ablative singular)
ympäristöiltäni (ablative plural)
ympäristölleni (allative singular)
ympäristöilleni (allative plural)
ympäristönäni (essive singular)
ympäristöinäni (essive plural)
ympäristökseni (singular translative)
ympäristöikseni (plural translative)
ympäristöttäni (abessive singular)
ympäristöittäni (abessive plural)
ympäristöineni (comitative plural)
ympäristösi (nominative singular)
ympäristösi (nominative plural)
ympäristösi (accusative nominative singular)
ympäristösi (accusative nominative plural)
ympäristösi (accusative genitive singular)
ympäristösi (accusative genitive plural)
ympäristösi (genitive singular)
ympäristöjesi (genitive plural)
ympäristöäsi (partitive singular)
ympäristöjäsi (partitive plural)
ympäristössäsi (inessive singular)
ympäristöissäsi (inessive plural)
ympäristöstäsi (elative singular)
ympäristöistäsi (elative plural)
ympäristöösi (illative singular)
ympäristöihisi (illative plural)
ympäristölläsi (adessive singular)
ympäristöilläsi (adessive plural)
ympäristöltäsi (ablative singular)
ympäristöiltäsi (ablative plural)
ympäristöllesi (allative singular)
ympäristöillesi (allative plural)
ympäristönäsi (essive singular)
ympäristöinäsi (essive plural)
ympäristöksesi (singular translative)
ympäristöiksesi (plural translative)
ympäristöttäsi (abessive singular)
ympäristöittäsi (abessive plural)
ympäristöinesi (comitative plural)
ympäristömme (nominative singular)
ympäristömme (nominative plural)
ympäristömme (accusative nominative singular)
ympäristömme (accusative nominative plural)
ympäristömme (accusative genitive singular)
ympäristömme (accusative genitive plural)
ympäristömme (genitive singular)
ympäristöjemme (genitive plural)
ympäristöämme (partitive singular)
ympäristöjämme (partitive plural)
ympäristössämme (inessive singular)
ympäristöissämme (inessive plural)
ympäristöstämme (elative singular)
ympäristöistämme (elative plural)
ympäristöömme (illative singular)
ympäristöihimme (illative plural)
ympäristöllämme (adessive singular)
ympäristöillämme (adessive plural)
ympäristöltämme (ablative singular)
ympäristöiltämme (ablative plural)
ympäristöllemme (allative singular)
ympäristöillemme (allative plural)
ympäristönämme (essive singular)
ympäristöinämme (essive plural)
ympäristöksemme (singular translative)
ympäristöiksemme (plural translative)
ympäristöttämme (abessive singular)
ympäristöittämme (abessive plural)
ympäristöinemme (comitative plural)
ympäristönne (nominative singular)
ympäristönne (nominative plural)
ympäristönne (accusative nominative singular)
ympäristönne (accusative nominative plural)
ympäristönne (accusative genitive singular)
ympäristönne (accusative genitive plural)
ympäristönne (genitive singular)
ympäristöjenne (genitive plural)
ympäristöänne (partitive singular)
ympäristöjänne (partitive plural)
ympäristössänne (inessive singular)
ympäristöissänne (inessive plural)
ympäristöstänne (elative singular)
ympäristöistänne (elative plural)
ympäristöönne (illative singular)
ympäristöihinne (illative plural)
ympäristöllänne (adessive singular)
ympäristöillänne (adessive plural)
ympäristöltänne (ablative singular)
ympäristöiltänne (ablative plural)
ympäristöllenne (allative singular)
ympäristöillenne (allative plural)
ympäristönänne (essive singular)
ympäristöinänne (essive plural)
ympäristöksenne (singular translative)
ympäristöiksenne (plural translative)
ympäristöttänne (abessive singular)
ympäristöittänne (abessive plural)
ympäristöinenne (comitative plural)
ympäristönsä (nominative singular)
ympäristönsä (nominative plural)
ympäristönsä (accusative nominative singular)
ympäristönsä (accusative nominative plural)
ympäristönsä (accusative genitive singular)
ympäristönsä (accusative genitive plural)
ympäristönsä (genitive singular)
ympäristöjensä (genitive plural)
ympäristöään (partitive singular)
ympäristöänsä (partitive singular)
ympäristöjään (partitive plural)
ympäristöjänsä (partitive plural)
ympäristössään (inessive singular)
ympäristössänsä (inessive singular)
ympäristöissään (inessive plural)
ympäristöissänsä (inessive plural)
ympäristöstään (elative singular)
ympäristöstänsä (elative singular)
ympäristöistään (elative plural)
ympäristöistänsä (elative plural)
ympäristöönsä (illative singular)
ympäristöihinsä (illative plural)
ympäristöllään (adessive singular)
ympäristöllänsä (adessive singular)
ympäristöillään (adessive plural)
ympäristöillänsä (adessive plural)
ympäristöltään (ablative singular)
ympäristöltänsä (ablative singular)
ympäristöiltään (ablative plural)
ympäristöiltänsä (ablative plural)
ympäristölleen (allative singular)
ympäristöllensä (allative singular)
ympäristöilleen (allative plural)
ympäristöillensä (allative plural)
ympäristönään (essive singular)
ympäristönänsä (essive singular)
ympäristöinään (essive plural)
ympäristöinänsä (essive plural)
ympäristökseen (singular translative)
ympäristöksensä (singular translative)
ympäristöikseen (plural translative)
ympäristöiksensä (plural translative)
ympäristöttään (abessive singular)
ympäristöttänsä (abessive singular)
ympäristöittään (abessive plural)
ympäristöittänsä (abessive plural)
ympäristöineen (comitative plural)
ympäristöinensä (comitative plural)


alue

alue (nominative singular)
alueet (nominative plural)
alue (accusative nominative singular)
alueet (accusative nominative plural)
alueen (accusative genitive singular)
alueet (accusative genitive plural)
alueen (genitive singular)
alueiden (genitive plural)
alueitten (genitive plural)
aluetta (partitive singular)
alueita (partitive plural)
alueessa (inessive singular)
alueissa (inessive plural)
alueesta (elative singular)
alueista (elative plural)
alueeseen (illative singular)
alueisiin (illative plural)
alueihin (illative plural)
alueella (adessive singular)
alueilla (adessive plural)
alueelta (ablative singular)
alueilta (ablative plural)
alueelle (allative singular)
alueille (allative plural)
alueena (essive singular)
alueina (essive plural)
alueeksi (singular translative)
alueiksi (plural translative)
alueetta (abessive singular)
alueitta (abessive plural)
aluein (instructive plural)
See the possessive forms below. (comitative)
alueeni (nominative singular)
alueeni (nominative plural)
alueeni (accusative nominative singular)
alueeni (accusative nominative plural)
alueeni (accusative genitive singular)
alueeni (accusative genitive plural)
alueeni (genitive singular)
alueideni (genitive plural)
alueitteni (genitive plural)
aluettani (partitive singular)
alueitani (partitive plural)
alueessani (inessive singular)
alueissani (inessive plural)
alueestani (elative singular)
alueistani (elative plural)
alueeseeni (illative singular)
alueisiini (illative plural)
alueihini (illative plural)
alueellani (adessive singular)
alueillani (adessive plural)
alueeltani (ablative singular)
alueiltani (ablative plural)
alueelleni (allative singular)
alueilleni (allative plural)
alueenani (essive singular)
alueinani (essive plural)
alueekseni (singular translative)
alueikseni (plural translative)
alueettani (abessive singular)
alueittani (abessive plural)
alueineni (comitative plural)
alueesi (nominative singular)
alueesi (nominative plural)
alueesi (accusative nominative singular)
alueesi (accusative nominative plural)
alueesi (accusative genitive singular)
alueesi (accusative genitive plural)
alueesi (genitive singular)
alueidesi (genitive plural)
alueittesi (genitive plural)
aluettasi (partitive singular)
alueitasi (partitive plural)
alueessasi (inessive singular)
alueissasi (inessive plural)
alueestasi (elative singular)
alueistasi (elative plural)
alueeseesi (illative singular)
alueisiisi (illative plural)
alueihisi (illative plural)
alueellasi (adessive singular)
alueillasi (adessive plural)
alueeltasi (ablative singular)
alueiltasi (ablative plural)
alueellesi (allative singular)
alueillesi (allative plural)
alueenasi (essive singular)
alueinasi (essive plural)
alueeksesi (singular translative)
alueiksesi (plural translative)
alueettasi (abessive singular)
alueittasi (abessive plural)
alueinesi (comitative plural)
alueemme (nominative singular)
alueemme (nominative plural)
alueemme (accusative nominative singular)
alueemme (accusative nominative plural)
alueemme (accusative genitive singular)
alueemme (accusative genitive plural)
alueemme (genitive singular)
alueidemme (genitive plural)
alueittemme (genitive plural)
aluettamme (partitive singular)
alueitamme (partitive plural)
alueessamme (inessive singular)
alueissamme (inessive plural)
alueestamme (elative singular)
alueistamme (elative plural)
alueeseemme (illative singular)
alueisiimme (illative plural)
alueihimme (illative plural)
alueellamme (adessive singular)
alueillamme (adessive plural)
alueeltamme (ablative singular)
alueiltamme (ablative plural)
alueellemme (allative singular)
alueillemme (allative plural)
alueenamme (essive singular)
alueinamme (essive plural)
alueeksemme (singular translative)
alueiksemme (plural translative)
alueettamme (abessive singular)
alueittamme (abessive plural)
alueinemme (comitative plural)
alueenne (nominative singular)
alueenne (nominative plural)
alueenne (accusative nominative singular)
alueenne (accusative nominative plural)
alueenne (accusative genitive singular)
alueenne (accusative genitive plural)
alueenne (genitive singular)
alueidenne (genitive plural)
alueittenne (genitive plural)
aluettanne (partitive singular)
alueitanne (partitive plural)
alueessanne (inessive singular)
alueissanne (inessive plural)
alueestanne (elative singular)
alueistanne (elative plural)
alueeseenne (illative singular)
alueisiinne (illative plural)
alueihinne (illative plural)
alueellanne (adessive singular)
alueillanne (adessive plural)
alueeltanne (ablative singular)
alueiltanne (ablative plural)
alueellenne (allative singular)
alueillenne (allative plural)
alueenanne (essive singular)
alueinanne (essive plural)
alueeksenne (singular translative)
alueiksenne (plural translative)
alueettanne (abessive singular)
alueittanne (abessive plural)
alueinenne (comitative plural)
alueensa (nominative singular)
alueensa (nominative plural)
alueensa (accusative nominative singular)
alueensa (accusative nominative plural)
alueensa (accusative genitive singular)
alueensa (accusative genitive plural)
alueensa (genitive singular)
alueidensa (genitive plural)
alueittensa (genitive plural)
aluettaan (partitive singular)
aluettansa (partitive singular)
alueitaan (partitive plural)
alueitansa (partitive plural)
alueessaan (inessive singular)
alueessansa (inessive singular)
alueissaan (inessive plural)
alueissansa (inessive plural)
alueestaan (elative singular)
alueestansa (elative singular)
alueistaan (elative plural)
alueistansa (elative plural)
alueeseensa (illative singular)
alueisiinsa (illative plural)
alueihinsa (illative plural)
alueellaan (adessive singular)
alueellansa (adessive singular)
alueillaan (adessive plural)
alueillansa (adessive plural)
alueeltaan (ablative singular)
alueeltansa (ablative singular)
alueiltaan (ablative plural)
alueiltansa (ablative plural)
alueelleen (allative singular)
alueellensa (allative singular)
alueilleen (allative plural)
alueillensa (allative plural)
alueenaan (essive singular)
alueenansa (essive singular)
alueinaan (essive plural)
alueinansa (essive plural)
alueekseen (singular translative)
alueeksensa (singular translative)
alueikseen (plural translative)
alueiksensa (plural translative)
alueettaan (abessive singular)
alueettansa (abessive singular)
alueittaan (abessive plural)
alueittansa (abessive plural)
alueineen (comitative plural)
alueinensa (comitative plural)

From Proto-Finnic *alovëh. Equivalent to ala + -ue. First attested in 1829 in an edition of the newspaper Oulun Wiikko-Sanomia. According to SSA, the form alue may have been adopted in the standard language as a result of Karelian influence (Karelian aloveh), being native only in the 'easternmost' of dialects; the Tavastian dialectal word aloo is probably related, however.


homology sphere

en-noun

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#95;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


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


fiber bundle

fiber bundles (plural)
fibre bundle (alternative)

Herbert Seifert
Norman Steenrod
Project Euclid
fiber bundle

(mathematics): 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#92;times#91;0,1#93; (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#92;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#95;#92;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).


Seifert fiber space

en-noun

Seifert fiber space (plural Seifert fiber spaces)

Seifert fiber spaces (plural)
Seifert fibered space (alternative)
Seifert fibre space (alternative)
Seifert fibred space (alternative)

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


Seifert fibered space

Seifert fibered space (plural Seifert fibered spaces)

Seifert fiber space
Seifert fiber space#English

(topology, US) Alternative spelling of Seifert fiber space.


Seifert fibre space

Seifert fibre space (plural Seifert fibre spaces)

(topology, British) Alternative spelling of Seifert fiber space.


Seifert fibred space

Seifert fibred space (plural Seifert fibred spaces)


second-countable

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


homoeomorphic

(British, topology) Alternative spelling of homeomorphic.


homoeomorphism

(British spelling, topology) Alternative spelling of homeomorphism.


topological group

topological group (plural topological groups)

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


T₂

International Union of Pure and Applied Chemistry

(topology) The Hausdorff property; a category of topological spaces with this property.


T₂

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


handlebody

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


top-dimensional

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


cube with handles

cube with handles (plural cubes with handles)

(topology) A three-dimensional handlebody


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 knot

(topology) A pretzel link which is a knot.


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.


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


yhtenäinen

fi-adj

yhtenäinen (comparative yhtenäisempi, superlative yhtenäisin)

yhtenäisempi (comparative)
yhtenäisin (superlative)
yhtenäinen (nominative singular)
yhtenäiset (nominative plural)
yhtenäinen (accusative nominative singular)
yhtenäiset (accusative nominative plural)
yhtenäisen (accusative genitive singular)
yhtenäiset (accusative genitive plural)
yhtenäisen (genitive singular)
yhtenäisten (genitive plural)
yhtenäisien (genitive plural)
yhtenäistä (partitive singular)
yhtenäisiä (partitive plural)
yhtenäisessä (inessive singular)
yhtenäisissä (inessive plural)
yhtenäisestä (elative singular)
yhtenäisistä (elative plural)
yhtenäiseen (illative singular)
yhtenäisiin (illative plural)
yhtenäisellä (adessive singular)
yhtenäisillä (adessive plural)
yhtenäiseltä (ablative singular)
yhtenäisiltä (ablative plural)
yhtenäiselle (allative singular)
yhtenäisille (allative plural)
yhtenäisenä (essive singular)
yhtenäisinä (essive plural)
yhtenäiseksi (singular translative)
yhtenäisiksi (plural translative)
yhtenäisettä (abessive singular)
yhtenäisittä (abessive plural)
yhtenäisin (instructive plural)
yhtenäisine (comitative plural)
yhtenäiseni (nominative singular)
yhtenäiseni (nominative plural)
yhtenäiseni (accusative nominative singular)
yhtenäiseni (accusative nominative plural)
yhtenäiseni (accusative genitive singular)
yhtenäiseni (accusative genitive plural)
yhtenäiseni (genitive singular)
yhtenäisteni (genitive plural)
yhtenäisieni (genitive plural)
yhtenäistäni (partitive singular)
yhtenäisiäni (partitive plural)
yhtenäisessäni (inessive singular)
yhtenäisissäni (inessive plural)
yhtenäisestäni (elative singular)
yhtenäisistäni (elative plural)
yhtenäiseeni (illative singular)
yhtenäisiini (illative plural)
yhtenäiselläni (adessive singular)
yhtenäisilläni (adessive plural)
yhtenäiseltäni (ablative singular)
yhtenäisiltäni (ablative plural)
yhtenäiselleni (allative singular)
yhtenäisilleni (allative plural)
yhtenäisenäni (essive singular)
yhtenäisinäni (essive plural)
yhtenäisekseni (singular translative)
yhtenäisikseni (plural translative)
yhtenäisettäni (abessive singular)
yhtenäisittäni (abessive plural)
yhtenäisineni (comitative plural)
yhtenäisesi (nominative singular)
yhtenäisesi (nominative plural)
yhtenäisesi (accusative nominative singular)
yhtenäisesi (accusative nominative plural)
yhtenäisesi (accusative genitive singular)
yhtenäisesi (accusative genitive plural)
yhtenäisesi (genitive singular)
yhtenäistesi (genitive plural)
yhtenäisiesi (genitive plural)
yhtenäistäsi (partitive singular)
yhtenäisiäsi (partitive plural)
yhtenäisessäsi (inessive singular)
yhtenäisissäsi (inessive plural)
yhtenäisestäsi (elative singular)
yhtenäisistäsi (elative plural)
yhtenäiseesi (illative singular)
yhtenäisiisi (illative plural)
yhtenäiselläsi (adessive singular)
yhtenäisilläsi (adessive plural)
yhtenäiseltäsi (ablative singular)
yhtenäisiltäsi (ablative plural)
yhtenäisellesi (allative singular)
yhtenäisillesi (allative plural)
yhtenäisenäsi (essive singular)
yhtenäisinäsi (essive plural)
yhtenäiseksesi (singular translative)
yhtenäisiksesi (plural translative)
yhtenäisettäsi (abessive singular)
yhtenäisittäsi (abessive plural)
yhtenäisinesi (comitative plural)
yhtenäisemme (nominative singular)
yhtenäisemme (nominative plural)
yhtenäisemme (accusative nominative singular)
yhtenäisemme (accusative nominative plural)
yhtenäisemme (accusative genitive singular)
yhtenäisemme (accusative genitive plural)
yhtenäisemme (genitive singular)
yhtenäistemme (genitive plural)
yhtenäisiemme (genitive plural)
yhtenäistämme (partitive singular)
yhtenäisiämme (partitive plural)
yhtenäisessämme (inessive singular)
yhtenäisissämme (inessive plural)
yhtenäisestämme (elative singular)
yhtenäisistämme (elative plural)
yhtenäiseemme (illative singular)
yhtenäisiimme (illative plural)
yhtenäisellämme (adessive singular)
yhtenäisillämme (adessive plural)
yhtenäiseltämme (ablative singular)
yhtenäisiltämme (ablative plural)
yhtenäisellemme (allative singular)
yhtenäisillemme (allative plural)
yhtenäisenämme (essive singular)
yhtenäisinämme (essive plural)
yhtenäiseksemme (singular translative)
yhtenäisiksemme (plural translative)
yhtenäisettämme (abessive singular)
yhtenäisittämme (abessive plural)
yhtenäisinemme (comitative plural)
yhtenäisenne (nominative singular)
yhtenäisenne (nominative plural)
yhtenäisenne (accusative nominative singular)
yhtenäisenne (accusative nominative plural)
yhtenäisenne (accusative genitive singular)
yhtenäisenne (accusative genitive plural)
yhtenäisenne (genitive singular)
yhtenäistenne (genitive plural)
yhtenäisienne (genitive plural)
yhtenäistänne (partitive singular)
yhtenäisiänne (partitive plural)
yhtenäisessänne (inessive singular)
yhtenäisissänne (inessive plural)
yhtenäisestänne (elative singular)
yhtenäisistänne (elative plural)
yhtenäiseenne (illative singular)
yhtenäisiinne (illative plural)
yhtenäisellänne (adessive singular)
yhtenäisillänne (adessive plural)
yhtenäiseltänne (ablative singular)
yhtenäisiltänne (ablative plural)
yhtenäisellenne (allative singular)
yhtenäisillenne (allative plural)
yhtenäisenänne (essive singular)
yhtenäisinänne (essive plural)
yhtenäiseksenne (singular translative)
yhtenäisiksenne (plural translative)
yhtenäisettänne (abessive singular)
yhtenäisittänne (abessive plural)
yhtenäisinenne (comitative plural)
yhtenäisensä (nominative singular)
yhtenäisensä (nominative plural)
yhtenäisensä (accusative nominative singular)
yhtenäisensä (accusative nominative plural)
yhtenäisensä (accusative genitive singular)
yhtenäisensä (accusative genitive plural)
yhtenäisensä (genitive singular)
yhtenäistensä (genitive plural)
yhtenäisiensä (genitive plural)
yhtenäistään (partitive singular)
yhtenäistänsä (partitive singular)
yhtenäisiään (partitive plural)
yhtenäisiänsä (partitive plural)
yhtenäisessään (inessive singular)
yhtenäisessänsä (inessive singular)
yhtenäisissään (inessive plural)
yhtenäisissänsä (inessive plural)
yhtenäisestään (elative singular)
yhtenäisestänsä (elative singular)
yhtenäisistään (elative plural)
yhtenäisistänsä (elative plural)
yhtenäiseensä (illative singular)
yhtenäisiinsä (illative plural)
yhtenäisellään (adessive singular)
yhtenäisellänsä (adessive singular)
yhtenäisillään (adessive plural)
yhtenäisillänsä (adessive plural)
yhtenäiseltään (ablative singular)
yhtenäiseltänsä (ablative singular)
yhtenäisiltään (ablative plural)
yhtenäisiltänsä (ablative plural)
yhtenäiselleen (allative singular)
yhtenäisellensä (allative singular)
yhtenäisilleen (allative plural)
yhtenäisillensä (allative plural)
yhtenäisenään (essive singular)
yhtenäisenänsä (essive singular)
yhtenäisinään (essive plural)
yhtenäisinänsä (essive plural)
yhtenäisekseen (singular translative)
yhtenäiseksensä (singular translative)
yhtenäisikseen (plural translative)
yhtenäisiksensä (plural translative)
yhtenäisettään (abessive singular)
yhtenäisettänsä (abessive singular)
yhtenäisittään (abessive plural)
yhtenäisittänsä (abessive plural)
yhtenäisineen (comitative plural)
yhtenäisinensä (comitative plural)

yhte- (“one”) + -näinen. Coined by Finnish explorer, historian and author Carl Axel Gottlund in 1828. Doublet with yksinäinen (“lonely”), which uses the strong stem.

coin

(mathematics, topology, graph theory) connected


Dehn surgery

en-noun

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.


solid torus

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


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


simple connectivity

(topology) The state of being simply connected.


cobertura

From Old Galician-Portuguese cobertura, from Late Latin coopertūra, from Latin coopertus.

(topology) cover (a set of sets whose union contains the given set)


aberto

Guinea-Bissau Creole: abertu

Indo-Portuguese: aberto

Kabuverdianu: abértu

Kristang: abertu

Principense: betu

Sãotomense: betu

From Old Galician-Portuguese aberto, from Latin apertus.


(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.


nullhomotopic

null-homotopic (alternative)
nulhomotopic (alternative)

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


nulhomotopic

null-homotopic (alternative)
nullhomotopic (alternative)


null-homotopic

nullhomotopic (alternative)
nulhomotopic (alternative)


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


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.


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.


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.


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.


ind

(winding number): Ultimately a clipping of Latin index.

(geometry, topology, mathematical analysis) winding number


stożek

stożek (nominative singular)
stożki (nominative plural)
stożka (genitive singular)
stożków (genitive plural)
stożkowi (dative singular)
stożkom (dative plural)
stożek (accusative singular)
stożki (accusative plural)
stożkiem (instrumental singular)
stożkami (instrumental plural)
stożku (locative singular)
stożkach (locative plural)
stożku (singular vocative)
stożki (plural vocative)

From stóg + -ek. Sense 2 is a semantic loan from English cone.


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.


closed set

(topology) A set whose complement is open.


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).


homologia

(topology) homology (system of quotient groups associated to a topological space)


homotopia

(topology) homotopy (continuous deformation of one continuous function to another)


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.


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.


vizinhança

/vɨ.zɨˈɲɐ̃.sɐ/

/vɨ.zɨˈɲɐ̃.sɐ/

From vizinho (“neighbour”) + -ança (“-ance”).


quotient space

Thus if #92;mathfrak#123;D#125; 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 #92;mathfrak#123;D#125; 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


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.


ℝ-order tree

ℝ-order trees (plural)
𝐑-order tree (alternative)

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


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.


vektoriavaruus

vektoriavaruus (nominative singular)
vektoriavaruudet (nominative plural)
vektoriavaruus (accusative nominative singular)
vektoriavaruudet (accusative nominative plural)
vektoriavaruuden (accusative genitive singular)
vektoriavaruudet (accusative genitive plural)
vektoriavaruuden (genitive singular)
vektoriavaruuksien (genitive plural)
vektoriavaruutta (partitive singular)
vektoriavaruuksia (partitive plural)
vektoriavaruudessa (inessive singular)
vektoriavaruuksissa (inessive plural)
vektoriavaruudesta (elative singular)
vektoriavaruuksista (elative plural)
vektoriavaruuteen (illative singular)
vektoriavaruuksiin (illative plural)
vektoriavaruudella (adessive singular)
vektoriavaruuksilla (adessive plural)
vektoriavaruudelta (ablative singular)
vektoriavaruuksilta (ablative plural)
vektoriavaruudelle (allative singular)
vektoriavaruuksille (allative plural)
vektoriavaruutena (essive singular)
vektoriavaruuksina (essive plural)
vektoriavaruudeksi (singular translative)
vektoriavaruuksiksi (plural translative)
vektoriavaruudetta (abessive singular)
vektoriavaruuksitta (abessive plural)
vektoriavaruuksin (instructive plural)
See the possessive forms below. (comitative)
vektoriavaruuteni (nominative singular)
vektoriavaruuteni (nominative plural)
vektoriavaruuteni (accusative nominative singular)
vektoriavaruuteni (accusative nominative plural)
vektoriavaruuteni (accusative genitive singular)
vektoriavaruuteni (accusative genitive plural)
vektoriavaruuteni (genitive singular)
vektoriavaruuksieni (genitive plural)
vektoriavaruuttani (partitive singular)
vektoriavaruuksiani (partitive plural)
vektoriavaruudessani (inessive singular)
vektoriavaruuksissani (inessive plural)
vektoriavaruudestani (elative singular)
vektoriavaruuksistani (elative plural)
vektoriavaruuteeni (illative singular)
vektoriavaruuksiini (illative plural)
vektoriavaruudellani (adessive singular)
vektoriavaruuksillani (adessive plural)
vektoriavaruudeltani (ablative singular)
vektoriavaruuksiltani (ablative plural)
vektoriavaruudelleni (allative singular)
vektoriavaruuksilleni (allative plural)
vektoriavaruutenani (essive singular)
vektoriavaruuksinani (essive plural)
vektoriavaruudekseni (singular translative)
vektoriavaruuksikseni (plural translative)
vektoriavaruudettani (abessive singular)
vektoriavaruuksittani (abessive plural)
vektoriavaruuksineni (comitative plural)
vektoriavaruutesi (nominative singular)
vektoriavaruutesi (nominative plural)
vektoriavaruutesi (accusative nominative singular)
vektoriavaruutesi (accusative nominative plural)
vektoriavaruutesi (accusative genitive singular)
vektoriavaruutesi (accusative genitive plural)
vektoriavaruutesi (genitive singular)
vektoriavaruuksiesi (genitive plural)
vektoriavaruuttasi (partitive singular)
vektoriavaruuksiasi (partitive plural)
vektoriavaruudessasi (inessive singular)
vektoriavaruuksissasi (inessive plural)
vektoriavaruudestasi (elative singular)
vektoriavaruuksistasi (elative plural)
vektoriavaruuteesi (illative singular)
vektoriavaruuksiisi (illative plural)
vektoriavaruudellasi (adessive singular)
vektoriavaruuksillasi (adessive plural)
vektoriavaruudeltasi (ablative singular)
vektoriavaruuksiltasi (ablative plural)
vektoriavaruudellesi (allative singular)
vektoriavaruuksillesi (allative plural)
vektoriavaruutenasi (essive singular)
vektoriavaruuksinasi (essive plural)
vektoriavaruudeksesi (singular translative)
vektoriavaruuksiksesi (plural translative)
vektoriavaruudettasi (abessive singular)
vektoriavaruuksittasi (abessive plural)
vektoriavaruuksinesi (comitative plural)
vektoriavaruutemme (nominative singular)
vektoriavaruutemme (nominative plural)
vektoriavaruutemme (accusative nominative singular)
vektoriavaruutemme (accusative nominative plural)
vektoriavaruutemme (accusative genitive singular)
vektoriavaruutemme (accusative genitive plural)
vektoriavaruutemme (genitive singular)
vektoriavaruuksiemme (genitive plural)
vektoriavaruuttamme (partitive singular)
vektoriavaruuksiamme (partitive plural)
vektoriavaruudessamme (inessive singular)
vektoriavaruuksissamme (inessive plural)
vektoriavaruudestamme (elative singular)
vektoriavaruuksistamme (elative plural)
vektoriavaruuteemme (illative singular)
vektoriavaruuksiimme (illative plural)
vektoriavaruudellamme (adessive singular)
vektoriavaruuksillamme (adessive plural)
vektoriavaruudeltamme (ablative singular)
vektoriavaruuksiltamme (ablative plural)
vektoriavaruudellemme (allative singular)
vektoriavaruuksillemme (allative plural)
vektoriavaruutenamme (essive singular)
vektoriavaruuksinamme (essive plural)
vektoriavaruudeksemme (singular translative)
vektoriavaruuksiksemme (plural translative)
vektoriavaruudettamme (abessive singular)
vektoriavaruuksittamme (abessive plural)
vektoriavaruuksinemme (comitative plural)
vektoriavaruutenne (nominative singular)
vektoriavaruutenne (nominative plural)
vektoriavaruutenne (accusative nominative singular)
vektoriavaruutenne (accusative nominative plural)
vektoriavaruutenne (accusative genitive singular)
vektoriavaruutenne (accusative genitive plural)
vektoriavaruutenne (genitive singular)
vektoriavaruuksienne (genitive plural)
vektoriavaruuttanne (partitive singular)
vektoriavaruuksianne (partitive plural)
vektoriavaruudessanne (inessive singular)
vektoriavaruuksissanne (inessive plural)
vektoriavaruudestanne (elative singular)
vektoriavaruuksistanne (elative plural)
vektoriavaruuteenne (illative singular)
vektoriavaruuksiinne (illative plural)
vektoriavaruudellanne (adessive singular)
vektoriavaruuksillanne (adessive plural)
vektoriavaruudeltanne (ablative singular)
vektoriavaruuksiltanne (ablative plural)
vektoriavaruudellenne (allative singular)
vektoriavaruuksillenne (allative plural)
vektoriavaruutenanne (essive singular)
vektoriavaruuksinanne (essive plural)
vektoriavaruudeksenne (singular translative)
vektoriavaruuksiksenne (plural translative)
vektoriavaruudettanne (abessive singular)
vektoriavaruuksittanne (abessive plural)
vektoriavaruuksinenne (comitative plural)
vektoriavaruutensa (nominative singular)
vektoriavaruutensa (nominative plural)
vektoriavaruutensa (accusative nominative singular)
vektoriavaruutensa (accusative nominative plural)
vektoriavaruutensa (accusative genitive singular)
vektoriavaruutensa (accusative genitive plural)
vektoriavaruutensa (genitive singular)
vektoriavaruuksiensa (genitive plural)
vektoriavaruuttaan (partitive singular)
vektoriavaruuttansa (partitive singular)
vektoriavaruuksiaan (partitive plural)
vektoriavaruuksiansa (partitive plural)
vektoriavaruudessaan (inessive singular)
vektoriavaruudessansa (inessive singular)
vektoriavaruuksissaan (inessive plural)
vektoriavaruuksissansa (inessive plural)
vektoriavaruudestaan (elative singular)
vektoriavaruudestansa (elative singular)
vektoriavaruuksistaan (elative plural)
vektoriavaruuksistansa (elative plural)
vektoriavaruuteensa (illative singular)
vektoriavaruuksiinsa (illative plural)
vektoriavaruudellaan (adessive singular)
vektoriavaruudellansa (adessive singular)
vektoriavaruuksillaan (adessive plural)
vektoriavaruuksillansa (adessive plural)
vektoriavaruudeltaan (ablative singular)
vektoriavaruudeltansa (ablative singular)
vektoriavaruuksiltaan (ablative plural)
vektoriavaruuksiltansa (ablative plural)
vektoriavaruudelleen (allative singular)
vektoriavaruudellensa (allative singular)
vektoriavaruuksilleen (allative plural)
vektoriavaruuksillensa (allative plural)
vektoriavaruutenaan (essive singular)
vektoriavaruutenansa (essive singular)
vektoriavaruuksinaan (essive plural)
vektoriavaruuksinansa (essive plural)
vektoriavaruudekseen (singular translative)
vektoriavaruudeksensa (singular translative)
vektoriavaruuksikseen (plural translative)
vektoriavaruuksiksensa (plural translative)
vektoriavaruudettaan (abessive singular)
vektoriavaruudettansa (abessive singular)
vektoriavaruuksittaan (abessive plural)
vektoriavaruuksittansa (abessive plural)
vektoriavaruuksineen (comitative plural)
vektoriavaruuksinensa (comitative plural)

[ˈʋe̞kt̪o̞riˌɑ̝ʋɑ̝ruːs̠]

(algebra, geometry, mathematics, topology) vector space


Hausdorff space

Hausdorff space (plural Hausdorff spaces)

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

named-after

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.


zwarty

pl-adj

zwarty (comparative bardziej zwarty, superlative najbardziej zwarty, derived adverb zwarto)

bardziej zwarty (comparative)
najbardziej zwarty (superlative)
zwarty (masculine nominative singular)
zwarta (feminine nominative singular)
zwarte (neuter nominative singular)
zwarci (nominative plural virile)
zwarte (nominative plural)
zwartego (genitive masculine singular)
zwartej (feminine genitive singular)
zwartego (genitive neuter singular)
zwartych (genitive plural)
zwartych (genitive plural virile)
zwartemu (dative masculine singular)
zwartej (dative feminine singular)
zwartemu (dative neuter singular)
zwartym (dative plural)
zwartym (dative plural virile)
zwartego (accusative animate masculine singular)
zwarty (accusative inanimate masculine singular)
zwartą (accusative feminine singular)
zwarte (accusative neuter singular)
zwartych (accusative plural virile)
zwarte (accusative plural)
zwartym (instrumental masculine singular)
zwartą (feminine instrumental singular)
zwartym (instrumental neuter singular)
zwartymi (instrumental plural)
zwartymi (instrumental plural virile)
zwartym (locative masculine singular)
zwartej (feminine locative singular)
zwartym (locative neuter singular)
zwartych (locative plural)
zwartych (locative plural virile)


composante

(mathematics, topology) (connected) component


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#92;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)

deformation retracts (plural)
deformation-retract (alternative)

(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.


paracompact

fr-adj

paracompact (feminine paracompacte, masculine plural paracompacts, feminine plural paracompactes)

paracompacte (feminine)
paracompacts (masculine plural)
paracompactes (feminine plural)


point-set topology

(mathematics, topology) The general field of topology, not restricting attention to specific classes of spaces, and not using algebraic topology.


open book

open book decomposition
open book decomposition

(mathematics, topology) An open book decomposition.


open book decomposition

en-noun

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.


samenhang

nl-noun

samenhang m (plural samenhangen, diminutive samenhangetje n)

samenhangen (plural)
samenhangetje (diminutive neuter)


samenhangendheid

samenhangendheid f (plural samenhangendheden)


cinquefoil

cinquefoils (plural)
cinqfoil (alternative)
quinquefoil (alternative)

/ˈsɪŋkˌfɔɪl/

/ˈsæŋkˌfɔɪl/

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.


hyrna

is-noun

hyrna f (genitive singular hyrnu, nominative plural hyrnur)

hyrnu (genitive singular)
hyrnur (nominative plural)
hyrna (indefinite nominative singular)
hyrnan (definite nominative singular)
hyrnur (indefinite nominative plural)
hyrnurnar (definite nominative plural)
hyrnu (accusative indefinite singular)
hyrnuna (accusative definite singular)
hyrnur (accusative indefinite plural)
hyrnurnar (accusative definite plural)
hyrnu (dative indefinite singular)
hyrnunni (dative definite singular)
hyrnum (dative indefinite plural)
hyrnunum (dative definite plural)
hyrnu (genitive indefinite singular)
hyrnunnar (definite genitive singular)
hyrna (genitive indefinite plural)
hyrnanna (definite genitive plural)

From Old Norse hyrna, from Proto-Germanic *hurnijǭ. Related to horn (“an angle, a corner”).


homotooppinen

homotooppinen (nominative singular)
homotooppiset (nominative plural)
homotooppinen (accusative nominative singular)
homotooppiset (accusative nominative plural)
homotooppisen (accusative genitive singular)
homotooppiset (accusative genitive plural)
homotooppisen (genitive singular)
homotooppisten (genitive plural)
homotooppisien (genitive plural)
homotooppista (partitive singular)
homotooppisia (partitive plural)
homotooppisessa (inessive singular)
homotooppisissa (inessive plural)
homotooppisesta (elative singular)
homotooppisista (elative plural)
homotooppiseen (illative singular)
homotooppisiin (illative plural)
homotooppisella (adessive singular)
homotooppisilla (adessive plural)
homotooppiselta (ablative singular)
homotooppisilta (ablative plural)
homotooppiselle (allative singular)
homotooppisille (allative plural)
homotooppisena (essive singular)
homotooppisina (essive plural)
homotooppiseksi (singular translative)
homotooppisiksi (plural translative)
homotooppisetta (abessive singular)
homotooppisitta (abessive plural)
homotooppisin (instructive plural)
homotooppisine (comitative plural)
homotooppiseni (nominative singular)
homotooppiseni (nominative plural)
homotooppiseni (accusative nominative singular)
homotooppiseni (accusative nominative plural)
homotooppiseni (accusative genitive singular)
homotooppiseni (accusative genitive plural)
homotooppiseni (genitive singular)
homotooppisteni (genitive plural)
homotooppisieni (genitive plural)
homotooppistani (partitive singular)
homotooppisiani (partitive plural)
homotooppisessani (inessive singular)
homotooppisissani (inessive plural)
homotooppisestani (elative singular)
homotooppisistani (elative plural)
homotooppiseeni (illative singular)
homotooppisiini (illative plural)
homotooppisellani (adessive singular)
homotooppisillani (adessive plural)
homotooppiseltani (ablative singular)
homotooppisiltani (ablative plural)
homotooppiselleni (allative singular)
homotooppisilleni (allative plural)
homotooppisenani (essive singular)
homotooppisinani (essive plural)
homotooppisekseni (singular translative)
homotooppisikseni (plural translative)
homotooppisettani (abessive singular)
homotooppisittani (abessive plural)
homotooppisineni (comitative plural)
homotooppisesi (nominative singular)
homotooppisesi (nominative plural)
homotooppisesi (accusative nominative singular)
homotooppisesi (accusative nominative plural)
homotooppisesi (accusative genitive singular)
homotooppisesi (accusative genitive plural)
homotooppisesi (genitive singular)
homotooppistesi (genitive plural)
homotooppisiesi (genitive plural)
homotooppistasi (partitive singular)
homotooppisiasi (partitive plural)
homotooppisessasi (inessive singular)
homotooppisissasi (inessive plural)
homotooppisestasi (elative singular)
homotooppisistasi (elative plural)
homotooppiseesi (illative singular)
homotooppisiisi (illative plural)
homotooppisellasi (adessive singular)
homotooppisillasi (adessive plural)
homotooppiseltasi (ablative singular)
homotooppisiltasi (ablative plural)
homotooppisellesi (allative singular)
homotooppisillesi (allative plural)
homotooppisenasi (essive singular)
homotooppisinasi (essive plural)
homotooppiseksesi (singular translative)
homotooppisiksesi (plural translative)
homotooppisettasi (abessive singular)
homotooppisittasi (abessive plural)
homotooppisinesi (comitative plural)
homotooppisemme (nominative singular)
homotooppisemme (nominative plural)
homotooppisemme (accusative nominative singular)
homotooppisemme (accusative nominative plural)
homotooppisemme (accusative genitive singular)
homotooppisemme (accusative genitive plural)
homotooppisemme (genitive singular)
homotooppistemme (genitive plural)
homotooppisiemme (genitive plural)
homotooppistamme (partitive singular)
homotooppisiamme (partitive plural)
homotooppisessamme (inessive singular)
homotooppisissamme (inessive plural)
homotooppisestamme (elative singular)
homotooppisistamme (elative plural)
homotooppiseemme (illative singular)
homotooppisiimme (illative plural)
homotooppisellamme (adessive singular)
homotooppisillamme (adessive plural)
homotooppiseltamme (ablative singular)
homotooppisiltamme (ablative plural)
homotooppisellemme (allative singular)
homotooppisillemme (allative plural)
homotooppisenamme (essive singular)
homotooppisinamme (essive plural)
homotooppiseksemme (singular translative)
homotooppisiksemme (plural translative)
homotooppisettamme (abessive singular)
homotooppisittamme (abessive plural)
homotooppisinemme (comitative plural)
homotooppisenne (nominative singular)
homotooppisenne (nominative plural)
homotooppisenne (accusative nominative singular)
homotooppisenne (accusative nominative plural)
homotooppisenne (accusative genitive singular)
homotooppisenne (accusative genitive plural)
homotooppisenne (genitive singular)
homotooppistenne (genitive plural)
homotooppisienne (genitive plural)
homotooppistanne (partitive singular)
homotooppisianne (partitive plural)
homotooppisessanne (inessive singular)
homotooppisissanne (inessive plural)
homotooppisestanne (elative singular)
homotooppisistanne (elative plural)
homotooppiseenne (illative singular)
homotooppisiinne (illative plural)
homotooppisellanne (adessive singular)
homotooppisillanne (adessive plural)
homotooppiseltanne (ablative singular)
homotooppisiltanne (ablative plural)
homotooppisellenne (allative singular)
homotooppisillenne (allative plural)
homotooppisenanne (essive singular)
homotooppisinanne (essive plural)
homotooppiseksenne (singular translative)
homotooppisiksenne (plural translative)
homotooppisettanne (abessive singular)
homotooppisittanne (abessive plural)
homotooppisinenne (comitative plural)
homotooppisensa (nominative singular)
homotooppisensa (nominative plural)
homotooppisensa (accusative nominative singular)
homotooppisensa (accusative nominative plural)
homotooppisensa (accusative genitive singular)
homotooppisensa (accusative genitive plural)
homotooppisensa (genitive singular)
homotooppistensa (genitive plural)
homotooppisiensa (genitive plural)
homotooppistaan (partitive singular)
homotooppistansa (partitive singular)
homotooppisiaan (partitive plural)
homotooppisiansa (partitive plural)
homotooppisessaan (inessive singular)
homotooppisessansa (inessive singular)
homotooppisissaan (inessive plural)
homotooppisissansa (inessive plural)
homotooppisestaan (elative singular)
homotooppisestansa (elative singular)
homotooppisistaan (elative plural)
homotooppisistansa (elative plural)
homotooppiseensa (illative singular)
homotooppisiinsa (illative plural)
homotooppisellaan (adessive singular)
homotooppisellansa (adessive singular)
homotooppisillaan (adessive plural)
homotooppisillansa (adessive plural)
homotooppiseltaan (ablative singular)
homotooppiseltansa (ablative singular)
homotooppisiltaan (ablative plural)
homotooppisiltansa (ablative plural)
homotooppiselleen (allative singular)
homotooppisellensa (allative singular)
homotooppisilleen (allative plural)
homotooppisillensa (allative plural)
homotooppisenaan (essive singular)
homotooppisenansa (essive singular)
homotooppisinaan (essive plural)
homotooppisinansa (essive plural)
homotooppisekseen (singular translative)
homotooppiseksensa (singular translative)
homotooppisikseen (plural translative)
homotooppisiksensa (plural translative)
homotooppisettaan (abessive singular)
homotooppisettansa (abessive singular)
homotooppisittaan (abessive plural)
homotooppisittansa (abessive plural)
homotooppisineen (comitative plural)
homotooppisinensa (comitative plural)

Internationalism (see English homotopic). Adapted with the adjectival suffix + -inen.


interior point

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


ℬ

(topology) A variable name for a Banach space.

(topology) A variable name for a basis of a topological space.


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.


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.


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.


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.


Fσ

From French fermé (“closed”) for the 'F' and French somme (“sum”) for the 'σ'. The 'σ' can be interpreted to mean "countable union."

(mathematical analysis, topology) A countable union of closed sets.


Gδ

From German Gebiet (“neighborhood”) for 'G' and German Durchschnitt (“intersection”) for the 'δ'.

(mathematical analysis, topology) A countable intersection of open sets.


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.


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.


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.


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.


hoiméamorfacht

head

hoiméamorfacht f (genitive singular hoiméamorfachta)

hoiméamorfachta (genitive singular)
hoiméamorfacht (indefinite nominative singular)
a hoiméamorfacht (indefinite singular vocative)
hoiméamorfachta (genitive indefinite singular)
hoiméamorfacht (dative indefinite singular)
an hoiméamorfacht (definite nominative singular)
na hoiméamorfachta (definite genitive singular)
leis an hoiméamorfacht (dative definite singular)
don hoiméamorfacht (dative definite singular)


hoiméamorfachas

hoiméamorfachas m (genitive singular hoiméamorfachais)

hoiméamorfachais (genitive singular)
hoiméamorfachas (indefinite nominative singular)
a hoiméamorfachais (indefinite singular vocative)
hoiméamorfachais (genitive indefinite singular)
hoiméamorfachas (dative indefinite singular)
an hoiméamorfachas (definite nominative singular)
an hoiméamorfachais (definite genitive singular)
leis an hoiméamorfachas (dative definite singular)
don hoiméamorfachas (dative definite singular)


retrarre

it-verb

retràrre (first-person singular present retràggo, first-person singular past historic retràssi, past participle retràtto, first-person singular imperfect retraévo, auxiliary avére)

retràrre (canonical)
retràggo (first-person present singular)
retràssi (first-person historic past singular)
retràtto (participle past)
retraévo (first-person imperfect singular)
avére (auxiliary)
retràrre (infinitive)
avére (auxiliary)
retraèndo (gerund)
retraènte (participle present)
retràtto (participle past)
retràggo (first-person indicative present singular)
retrài (indicative present second-person singular)
retràe (indicative present singular third-person)
retraiàmo (first-person indicative plural present)
retraéte (indicative plural present second-person)
retràggono (indicative plural present third-person)
retraévo (first-person imperfect indicative singular)
retraévi (imperfect indicative second-person singular)
retraéva (imperfect indicative singular third-person)
retraevàmo (first-person imperfect indicative plural)
retraevàte (imperfect indicative plural second-person)
retraévano (imperfect indicative plural third-person)
retràssi (first-person historic indicative past singular)
retraésti (historic indicative past second-person singular)
retràsse (historic indicative past singular third-person)
retraémmo (first-person historic indicative past plural)
retraéste (historic indicative past plural second-person)
retràssero (historic indicative past plural third-person)
retrarrò (first-person future indicative singular)
retrarrài (future indicative second-person singular)
retrarrà (future indicative singular third-person)
retrarrémo (first-person future indicative plural)
retrarréte (future indicative plural second-person)
retrarrànno (future indicative plural third-person)
retrarrèi (conditional first-person singular)
retrarrésti (conditional second-person singular)
retrarrèbbe (conditional singular third-person)
retrarrébbe (conditional singular third-person)
retrarrémmo (conditional first-person plural)
retrarréste (conditional plural second-person)
retrarrèbbero (conditional plural third-person)
retrarrébbero (conditional plural third-person)
retràgga (first-person present singular subjunctive)
retràgga (present second-person singular subjunctive)
retràgga (present singular subjunctive third-person)
retraiàmo (first-person plural present subjunctive)
retraiàte (plural present second-person subjunctive)
retràggano (plural present subjunctive third-person)
retraéssi (first-person imperfect singular subjunctive)
retraéssi (imperfect second-person singular subjunctive)
retraésse (imperfect singular subjunctive third-person)
retraéssimo (first-person imperfect plural subjunctive)
retraéste (imperfect plural second-person subjunctive)
retraéssero (imperfect plural subjunctive third-person)
retrài (imperative second-person singular)
retràgga (formal imperative second-person-semantically singular third-person)
retraiàmo (first-person imperative plural)
retraéte (imperative plural second-person)
retràggano (formal imperative plural second-person-semantically third-person)
retràrre (imperative negative second-person singular)
retràgga (formal imperative negative second-person-semantically singular third-person)
retraiàmo (first-person imperative negative plural)
retraéte (imperative negative plural second-person)
retràggano (formal imperative negative plural second-person-semantically third-person)

(topology) to construct a retraction of (a topological space)


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.


preregular

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


jaðar

is-noun

jaðar m (genitive singular jaðars, nominative plural jaðrar)

jaðars (genitive singular)
jaðrar (nominative plural)
jaðar (indefinite nominative singular)
jaðarinn (definite nominative singular)
jaðrar (indefinite nominative plural)
jaðrarnir (definite nominative plural)
jaðar (accusative indefinite singular)
jaðarinn (accusative definite singular)
jaðra (accusative indefinite plural)
jaðrana (accusative definite plural)
jaðri (dative indefinite singular)
jaðrinum (dative definite singular)
jöðrum (dative indefinite plural)
jöðrunum (dative definite plural)
jaðars (genitive indefinite singular)
jaðarsins (definite genitive singular)
jaðra (genitive indefinite plural)
jaðranna (definite genitive plural)

From Old Norse jaðarr, from Proto-Germanic *edaraz.


zusammenhängend

de-adj

zusammenhängend (strong nominative masculine singular zusammenhängender, not comparable)

zusammenhängender (masculine nominative singular strong)
zusammenhängend (masculine predicative singular)
zusammenhängend (feminine predicative singular)
zusammenhängend (neuter predicative singular)
zusammenhängend (plural predicative)
zusammenhängender (masculine nominative singular strong without-article)
zusammenhängende (feminine nominative singular strong without-article)
zusammenhängendes (neuter nominative singular strong without-article)
zusammenhängende (nominative plural strong without-article)
zusammenhängenden (genitive masculine singular strong without-article)
zusammenhängender (feminine genitive singular strong without-article)
zusammenhängenden (genitive neuter singular strong without-article)
zusammenhängender (genitive plural strong without-article)
zusammenhängendem (dative masculine singular strong without-article)
zusammenhängender (dative feminine singular strong without-article)
zusammenhängendem (dative neuter singular strong without-article)
zusammenhängenden (dative plural strong without-article)
zusammenhängenden (accusative masculine singular strong without-article)
zusammenhängende (accusative feminine singular strong without-article)
zusammenhängendes (accusative neuter singular strong without-article)
zusammenhängende (accusative plural strong without-article)
der zusammenhängende (definite includes-article masculine nominative singular weak)
die zusammenhängende (definite feminine includes-article nominative singular weak)
das zusammenhängende (definite includes-article neuter nominative singular weak)
die zusammenhängenden (definite includes-article nominative plural weak)
des zusammenhängenden (definite genitive includes-article masculine singular weak)
der zusammenhängenden (definite feminine genitive includes-article singular weak)
des zusammenhängenden (definite genitive includes-article neuter singular weak)
der zusammenhängenden (definite genitive includes-article plural weak)
dem zusammenhängenden (dative definite includes-article masculine singular weak)
der zusammenhängenden (dative definite feminine includes-article singular weak)
dem zusammenhängenden (dative definite includes-article neuter singular weak)
den zusammenhängenden (dative definite includes-article plural weak)
den zusammenhängenden (accusative definite includes-article masculine singular weak)
die zusammenhängende (accusative definite feminine includes-article singular weak)
das zusammenhängende (accusative definite includes-article neuter singular weak)
die zusammenhängenden (accusative definite includes-article plural weak)
ein zusammenhängender (includes-article indefinite masculine mixed nominative singular)
eine zusammenhängende (feminine includes-article indefinite mixed nominative singular)
ein zusammenhängendes (includes-article indefinite mixed neuter nominative singular)
zusammenhängenden (indefinite mixed nominative plural)
keine zusammenhängenden (includes-article indefinite mixed negative nominative plural)
eines zusammenhängenden (genitive includes-article indefinite masculine mixed singular)
einer zusammenhängenden (feminine genitive includes-article indefinite mixed singular)
eines zusammenhängenden (genitive includes-article indefinite mixed neuter singular)
zusammenhängenden (genitive indefinite mixed plural)
keiner zusammenhängenden (genitive includes-article indefinite mixed negative plural)
einem zusammenhängenden (dative includes-article indefinite masculine mixed singular)
einer zusammenhängenden (dative feminine includes-article indefinite mixed singular)
einem zusammenhängenden (dative includes-article indefinite mixed neuter singular)
zusammenhängenden (dative indefinite mixed plural)
keinen zusammenhängenden (dative includes-article indefinite mixed negative plural)
einen zusammenhängenden (accusative includes-article indefinite masculine mixed singular)
eine zusammenhängende (accusative feminine includes-article indefinite mixed singular)
ein zusammenhängendes (accusative includes-article indefinite mixed neuter singular)
zusammenhängenden (accusative indefinite mixed plural)
keine zusammenhängenden (accusative includes-article indefinite mixed negative plural)

zusammenhängender Raum


spójny

pl-adj

spójny (comparative bardziej spójny, superlative najbardziej spójny, derived adverb spójnie)

bardziej spójny (comparative)
najbardziej spójny (superlative)
spójny (masculine nominative singular)
spójna (feminine nominative singular)
spójne (neuter nominative singular)
spójni (nominative plural virile)
spójne (nominative plural)
spójnego (genitive masculine singular)
spójnej (feminine genitive singular)
spójnego (genitive neuter singular)
spójnych (genitive plural)
spójnych (genitive plural virile)
spójnemu (dative masculine singular)
spójnej (dative feminine singular)
spójnemu (dative neuter singular)
spójnym (dative plural)
spójnym (dative plural virile)
spójnego (accusative animate masculine singular)
spójny (accusative inanimate masculine singular)
spójną (accusative feminine singular)
spójne (accusative neuter singular)
spójnych (accusative plural virile)
spójne (accusative plural)
spójnym (instrumental masculine singular)
spójną (feminine instrumental singular)
spójnym (instrumental neuter singular)
spójnymi (instrumental plural)
spójnymi (instrumental plural virile)
spójnym (locative masculine singular)
spójnej (feminine locative singular)
spójnym (locative neuter singular)
spójnych (locative plural)
spójnych (locative plural virile)


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.


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


Cauchy space

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


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


rączka

rączka (nominative singular)
rączki (nominative plural)
rączki (genitive singular)
rączek (genitive plural)
rączce (dative singular)
rączkom (dative plural)
rączkę (accusative singular)
rączki (accusative plural)
rączką (instrumental singular)
rączkami (instrumental plural)
rączce (locative singular)
rączkach (locative plural)
rączko (singular vocative)
rączki (plural vocative)


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.


單純

/tän⁵⁵ ʈ͡ʂʰu̯ən³⁵/

/tän⁵⁵ ʈ͡ʂʰu̯ən³⁵/

(geometry, algebraic topology) simplicial


bania

bańka (diminutive)
bania (nominative singular)
banie (nominative plural)
bani (genitive singular)
bani (genitive plural)
bań (genitive plural)
bani (dative singular)
baniom (dative plural)
banię (accusative singular)
banie (accusative plural)
banią (instrumental singular)
baniami (instrumental plural)
bani (locative singular)
baniach (locative plural)
banio (singular vocative)
banie (plural vocative)

(Middle Polish, mathematics, geometry, topology) ball


jednospójny

jednospójny (not comparable, no derived adverb)

jednospójny (masculine nominative singular)
jednospójna (feminine nominative singular)
jednospójne (neuter nominative singular)
jednospójni (nominative plural virile)
jednospójne (nominative plural)
jednospójnego (genitive masculine singular)
jednospójnej (feminine genitive singular)
jednospójnego (genitive neuter singular)
jednospójnych (genitive plural)
jednospójnych (genitive plural virile)
jednospójnemu (dative masculine singular)
jednospójnej (dative feminine singular)
jednospójnemu (dative neuter singular)
jednospójnym (dative plural)
jednospójnym (dative plural virile)
jednospójnego (accusative animate masculine singular)
jednospójny (accusative inanimate masculine singular)
jednospójną (accusative feminine singular)
jednospójne (accusative neuter singular)
jednospójnych (accusative plural virile)
jednospójne (accusative plural)
jednospójnym (instrumental masculine singular)
jednospójną (feminine instrumental singular)
jednospójnym (instrumental neuter singular)
jednospójnymi (instrumental plural)
jednospójnymi (instrumental plural virile)
jednospójnym (locative masculine singular)
jednospójnej (feminine locative singular)
jednospójnym (locative neuter singular)
jednospójnych (locative plural)
jednospójnych (locative plural virile)


Riemann sphere

Named after German mathematician Bernhard Riemann.

We use #92;hat#123;#92;Complex#125;#92;times (resp. #92;Complex) to denote the subset #92;#123;z#92;in#92;hat#92;Complex#58;z#92;ne 0#92;#125; (resp. #92;#123;z#92;in#92;hat#92;Complex#58;z#92;ne#92;infty#92;#125;) in the Riemann sphere #92;hat#92;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#47;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.