Topological property

Last updated June 19, 2023

In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

Properties of topological properties

A property $P$ is:

Hereditary, if for every topological space $(X,{\mathcal {T}})$ and subset $S\subseteq X,$ the subspace $\left(S,{\mathcal {T}}|_{S}\right)$ has property $P.$
Weakly hereditary, if for every topological space $(X,{\mathcal {T}})$ and closed subset $S\subseteq X,$ the subspace $\left(S,{\mathcal {T}}|_{S}\right)$ has property $P.$

Common topological properties

Cardinal functions

The cardinality |X| of the space X.
The cardinality $\vert$ τ(X) $\vert$ of the topology (the set of open subsets) of the space X.
Weightw(X), the least cardinality of a basis of the topology of the space X.
Densityd(X), the least cardinality of a subset of X whose closure is X.

Separation

Note that some of these terms are defined differently in older mathematical literature; see history of the separation axioms.

T₀ or Kolmogorov. A space is Kolmogorov if for every pair of distinct points x and y in the space, there is at least either an open set containing x but not y, or an open set containing y but not x.
T₁ or Fréchet. A space is Fréchet if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with T₀; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T₁ if all its singletons are closed. T₁ spaces are always T₀.
Sober. A space is sober if every irreducible closed set C has a unique generic point p. In other words, if C is not the (possibly nondisjoint) union of two smaller closed non-empty subsets, then there is a p such that the closure of {p} equals C, and p is the only point with this property.
T₂ or Hausdorff. A space is Hausdorff if every two distinct points have disjoint neighbourhoods. T₂ spaces are always T₁.
T_2½ or Urysohn. A space is Urysohn if every two distinct points have disjoint closed neighbourhoods. T_2½ spaces are always T₂.
Completely T₂ or completely Hausdorff. A space is completely T₂ if every two distinct points are separated by a function. Every completely Hausdorff space is Urysohn.
Regular. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.
T₃ or Regular Hausdorff. A space is regular Hausdorff if it is a regular T₀ space. (A regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.)
Completely regular. A space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are separated by a function.
T_3½, Tychonoff, Completely regular Hausdorff or Completely T₃. A Tychonoff space is a completely regular T₀ space. (A completely regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.
Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity.
T₄ or Normal Hausdorff. A normal space is Hausdorff if and only if it is T₁. Normal Hausdorff spaces are always Tychonoff.
Completely normal. A space is completely normal if any two separated sets have disjoint neighbourhoods.
T₅ or Completely normal Hausdorff. A completely normal space is Hausdorff if and only if it is T₁. Completely normal Hausdorff spaces are always normal Hausdorff.
Perfectly normal. A space is perfectly normal if any two disjoint closed sets are precisely separated by a function. A perfectly normal space must also be completely normal.
T₆ or Perfectly normal Hausdorff, or perfectly T₄. A space is perfectly normal Hausdorff, if it is both perfectly normal and T₁. A perfectly normal Hausdorff space must also be completely normal Hausdorff.
Discrete space. A space is discrete if all of its points are completely isolated, i.e. if any subset is open.
Number of isolated points. The number of isolated points of a topological space.

Countability conditions

Separable. A space is separable if it has a countable dense subset.
First-countable. A space is first-countable if every point has a countable local base.
Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.

Connectedness

Connected. A space is connected if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only clopen sets are the empty set and itself.
Locally connected. A space is locally connected if every point has a local base consisting of connected sets.
Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point.
Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Path-connected spaces are always connected.
Locally path-connected. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.
Arc-connected. A space X is arc-connected if for every two points x, y in X, there is an arc f from x to y, i.e., an injective continuous map f: [0,1] → X with p(0) = x and p(1) = y. Arc-connected spaces are path-connected.
Simply connected. A space X is simply connected if it is path-connected and every continuous map f: S¹ → X is homotopic to a constant map.
Locally simply connected. A space X is locally simply connected if every point x in X has a local base of neighborhoods U that is simply connected.
Semi-locally simply connected. A space X is semi-locally simply connected if every point has a local base of neighborhoods U such that every loop in U is contractible in X. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a universal cover.
Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected.
Hyperconnected. A space is hyperconnected if no two non-empty open sets are disjoint. Every hyperconnected space is connected.
Ultraconnected. A space is ultraconnected if no two non-empty closed sets are disjoint. Every ultraconnected space is path-connected.
Indiscrete or trivial. A space is indiscrete if the only open sets are the empty set and itself. Such a space is said to have the trivial topology.

Compactness

Compact. A space is compact if every open cover has a finite subcover. Some authors call these spaces quasicompact and reserve compact for Hausdorff spaces where every open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.
Sequentially compact. A space is sequentially compact if every sequence has a convergent subsequence.
Countably compact. A space is countably compact if every countable open cover has a finite subcover.
Pseudocompact. A space is pseudocompact if every continuous real-valued function on the space is bounded.
σ-compact. A space is σ-compact if it is the union of countably many compact subsets.
Lindelöf. A space is Lindelöf if every open cover has a countable subcover.
Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.
Locally compact. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Slightly different definitions are also used. Locally compact Hausdorff spaces are always Tychonoff.
Ultraconnected compact. In an ultra-connected compact space X every open cover must contain X itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a monolith.

Metrizability

Metrizable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable. Moreover, a topological space (X,T) is said to be metrizable if there exists a metric for X such that the metric topology T(d) is identical with the topology T.
Polish. A space is called Polish if it is metrizable with a separable and complete metric.
Locally metrizable. A space is locally metrizable if every point has a metrizable neighbourhood.

Miscellaneous

Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense.
Door space. A topological space is a door space if every subset is open or closed (or both).
Topological Homogeneity. A space X is (topologically) homogeneous if for every x and y in X there is a homeomorphism $f:X\to X$ such that $f(x)=y.$ Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous.
Finitely generated or Alexandrov. A space X is Alexandrov if arbitrary intersections of open sets in X are open, or equivalently if arbitrary unions of closed sets are closed. These are precisely the finitely generated members of the category of topological spaces and continuous maps.
Zero-dimensional. A space is zero-dimensional if it has a base of clopen sets. These are precisely the spaces with a small inductive dimension of 0.
Almost discrete. A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
Boolean. A space is Boolean if it is zero-dimensional, compact and Hausdorff (equivalently, totally disconnected, compact and Hausdorff). These are precisely the spaces that are homeomorphic to the Stone spaces of Boolean algebras.
Reidemeister torsion
$\kappa$ -resolvable. A space is said to be κ-resolvable^[1] (respectively: almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (respectively: almost disjoint over the ideal of nowhere dense subsets). If the space is not $\kappa$ -resolvable then it is called $\kappa$ -irresolvable.
Maximally resolvable. Space $X$ is maximally resolvable if it is $\Delta (X)$ -resolvable, where $\Delta (X)=\min\{|G|:G\neq \varnothing ,G{\mbox{ is open}}\}.$ Number $\Delta (X)$ is called dispersion character of $X.$
Strongly discrete. Set $D$ is strongly discrete subset of the space $X$ if the points in $D$ may be separated by pairwise disjoint neighborhoods. Space $X$ is said to be strongly discrete if every non-isolated point of $X$ is the accumulation point of some strongly discrete set.

Non-topological properties

There are many examples of properties of metric spaces, etc, which are not topological properties. To show a property $P$ is not topological, it is sufficient to find two homeomorphic topological spaces $X\cong Y$ such that $X$ has $P$ , but $Y$ does not have $P$ .

For example, the metric space properties of boundedness and completeness are not topological properties. Let $X=\mathbb {R}$ and $Y=(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}})$ be metric spaces with the standard metric. Then, $X\cong Y$ via the homeomorphism $\operatorname {arctan} \colon X\to Y$ . However, $X$ is complete but not bounded, while $Y$ is bounded but not complete.

Citations

↑ Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán (2008). "Resolvability and monotone normality". Israel Journal of Mathematics . 166 (1): 1–16. arXiv: math/0609092 . doi: 10.1007/s11856-008-1017-y . ISSN 0021-2172. S2CID 14743623.

Related Research Articles

<span class="mw-page-title-main">Compact space</span> Type of mathematical space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers $is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers is not compact either, because it excludes the two limiting values and . However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.$

In topology and related branches of mathematics, a Hausdorff space ( HOWS-dorf, HOWZ-dorf), separated space or T₂ space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T₂) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space $is said to be metrizable if there is a metric such that the topology induced by is Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.$

In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T₄: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T₄ space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T₅ spaces, and perfectly normal Hausdorff spaces, or T₆ spaces.

In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood.

In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff. More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space.

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space $is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the second axiom of countability . Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.$

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology.

In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.

In mathematics, a topological space $is called collectionwise normal if for every discrete family F i (i \in I) of closed subsets of there exists a pairwise disjoint family of open sets U i (i \in I), such that F i \subseteq U i . Here a family of subsets of is called discrete when every point of has a neighbourhood that intersects at most one of the sets from . An equivalent definition of collectionwise normal demands that the above U i (i \in I) themselves form a discrete family, which is stronger than pairwise disjoint.$

In mathematics, particularly topology, a G_δ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A G_δ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal G_δ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.

In general topology, a branch of mathematics, the Appert topology, named for Antoine Appert (1934), is a topology on the set X = {1, 2, 3, ...} of positive integers. In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer. The space X with the Appert topology is called the Appert space.

References

Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. p. 369. ISBN 9780486434797.
Munkres, James R. (2000). Topology. Prentice-Hall. ISBN 0-13-181629-2.

[2] Simon Moulieras, Maciej Lewenstein and Graciana Puentes, Entanglement engineering and topological protection by discrete-time quantum walks, Journal of Physics B: Atomic, Molecular and Optical Physics 46 (10), 104005 (2013). https://iopscience.iop.org/article/10.1088/0953-4075/46/10/104005/pdf

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán (2008). "Resolvability and monotone normality". Israel Journal of Mathematics . 166 (1): 1–16. arXiv: math/0609092 . doi: 10.1007/s11856-008-1017-y . ISSN 0021-2172. S2CID 14743623.

[1]