path

From Middle English path, peth, from Old English pæþ (“path, track”), from Proto-West Germanic *paþ, from Proto-Germanic *paþaz (“path”). The Proto-Germanic term is possibly borrowed from Iranian, from Proto-Iranian *pántaHh, from Proto-Indo-Iranian *pántaHs, from Proto-Indo-European *póntoh₁s, from the root *pent- (“to pass”), however this is disputed. Cognate with West Frisian paad, Dutch pad, German Pfad. Indo-Iranian cognates could be Avestan 𐬞𐬀𐬧𐬙𐬃 (paṇtā̊, “way”), Old Persian 𐎱𐎰 (p-θ /paθi/)), Sanskrit पन्था (panthā). See also English find). Doublet of panth.

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


suspension

en-noun

suspension (countable and uncountable, plural suspensions)

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

der

To get an intuitive feeling for the characteristics of H'-spaces, it is instructive to consider an important class of such spaces, the suspensions. The suspension of an arbitrary topological space Y is defined to be the quotient space of Y#92;timesI where Y#92;times 0 is identified to one point and Y#92;times 1 is identified to another point. For example, the suspension of a circle is a cylinder with the two ends collapsed into one point each; in other words, a space homeomorphic to a sphere.

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

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

suspensie

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


metric space

mathematical analysis
mathematical analysis

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


topological space

en-noun

topological space (plural topological spaces)

(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.


hyperbolic plane

hyperbolic plane (plural hyperbolic planes)

(mathematics) A topological space homeomorphic to a plane (or open disk) and equipped with a hyperbolic metric.


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


conifold

(mathematics) A certain generalization of a manifold, capable of containing conical singularities.


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.


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.


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.


Hilbert cube

Named after German mathematician David Hilbert (1862–1943)

named-after

topological space
topological space#English

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


Tychonoff cube

Named after Russian mathematician Андре́й Никола́евич Ти́хонов (Andréj Nikolájevič Tíxonov) (1906–1993).

lang

Андре́й Никола́евич Ти́хонов

Андре́й Никола́евич Ти́хонов

xlit

Андре́й Никола́евич Ти́хонов

Андре́й Никола́евич Ти́хонов

named-after

Андре́й Никола́евич Ти́хонов

A topological space which is a product of infinitely many closed intervals.


pseudomanifold

A special type of topological space that looks like a manifold at most of its points, but may contain singularities.


Teichmüller space

en-noun

Teichmüller space (plural Teichmüller spaces)

(geometry) A space T(S) assigned to a (real) topological (or differential) surface S: it parameterizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.


sober space

For a reader more interested in function spaces than in functors, this concludes the description of content except to add that the classification of coadjoint G’s by sets B bearing a topological topology relativizes to T₀ spaces. For other readers: and trivially to sober spaces.

1975, Mathematica Scandinavica, Societates Mathematicae, page 318:

1983, Houston Journal of Mathematics, Volume 9, University of Houston, page 192,
In the Hofmann and Lawson paper, it is proved that the topological space Spec(L) is a locally quasicompact sober space […] .

Thus sober spaces are necessarily T₀.

2002, P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, volume 2, Oxford University Press, page 492:

How rich are sober and L-sober spaces, are there important examples? It is well-known [25] that Hausdorff spaces, and hence compact T₀ (and so finite T₀ spaces), are sober spaces in the traditional setting. Furthermore, the soberification of a space—the spectrum of the topology of a space—is always sober; and if the original space is not Hausdorff, then its soberification is a sober space which is not Hausdorff. So there are many non-Hausdorff sober spaces as well.

2003, A. Pultr, S. E. Rodabaugh, “Chapter 6: Lattice-Valued Frames, Functor Categories, And Classes of Sober Spaces”, in Stephen Ernest Rodabaugh, Erich Peter Klement, editors, Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Springer, page 155:

When X is a noetherian sober space the construction of loc. cit. can be extended to an arbitrary lower-semicontinuous function for the constructible topology.

2004, Ofer Gabber, “Notes on some t-structures”, in Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, François Loeser, editors, Geometric Aspects of Dwork Theory, volume 1, Walter de Gruyter, page 711:

(topology) A topological space of which every join-irreducible closed subset is the closure of exactly one point of the space.


mathematical space

mathematical space (plural mathematical spaces)

(topology) A set (a "universe") that consists of selected mathematical objects that are treated as points, and selected relationships between them.

mathematischer Raum

a set that consists of selected mathematical object