In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant 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.

- Common topological properties
- Cardinal functions
- Separation
- Countability conditions
- Connectedness
- Compactness
- Metrizability
- Miscellaneous
- Non-topological properties
- See also
- References
- Bibliography

A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are *not* homeomorphic, it is sufficient to find a topological property which is not shared by them.

- The cardinality |
*X*| of the space*X*. - The cardinality
*τ*(*X*) of the topology of the space*X*. *Weight**w*(*X*), the least cardinality of a basis of the topology of the space*X*.*Density**d*(*X*), the least cardinality of a subset of*X*whose closure is*X*.

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

**T**or_{0}**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*.**T**or_{1}**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 T_{0}; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T_{1}if all its singletons are closed. T_{1}spaces are always T_{0}.**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 subsets, then there is a*p*such that the closure of {*p*} equals*C*, and*p*is the only point with this property.**T**or_{2}**Hausdorff**. A space is Hausdorff if every two distinct points have disjoint neighbourhoods. T_{2}spaces are always T_{1}.**T**or_{2½}**Urysohn**. A space is Urysohn if every two distinct points have disjoint*closed*neighbourhoods. T_{2½}spaces are always T_{2}.**Completely T**or_{2}**completely Hausdorff**. A space is completely T_{2}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.**T**or_{3}**Regular Hausdorff**. A space is regular Hausdorff if it is a regular T_{0}space. (A regular space is Hausdorff if and only if it is T_{0}, 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 T**. A Tychonoff space is a completely regular T_{3}_{0}space. (A completely regular space is Hausdorff if and only if it is T_{0}, 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.**T**or_{4}**Normal Hausdorff**. A normal space is Hausdorff if and only if it is T_{1}. Normal Hausdorff spaces are always Tychonoff.**Completely normal**. A space is completely normal if any two separated sets have disjoint neighbourhoods.**T**or_{5}**Completely normal Hausdorff**. A completely normal space is Hausdorff if and only if it is T_{1}. 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.**T**or_{6}**Perfectly normal Hausdorff**, or**perfectly T**. A space is perfectly normal Hausdorff, if it is both perfectly normal and T_{4}_{1}. 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.

**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.

**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*: S^{1}→*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.

**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**.

**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.

**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.**Topological Homogeneity**. A space*X*is (topologically) homogeneous if for every*x*and*y*in*X*there is a homeomorphism*f*:*X*→*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****-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 -resolvable then it is called -irresolvable.**Maximally resolvable**. Space is maximally resolvable if it is -resolvable, where . Number is called dispersion character of .**Strongly discrete**. Set is strongly discrete subset of the space if the points in may be separated by pairwise disjoint neighborhoods. Space is said to be strongly discrete if every non-isolated point of is the accumulation point of some strongly discrete set.

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

For example, the metric space properties of boundedness and completeness are not topological properties. Let and be metric spaces with the standard metric. Then, via the homeomorphism . However, is complete but not bounded, while is bounded but not complete.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

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

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 obvious, 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.

In topology and related branches of mathematics, **Tychonoff spaces** and **completely regular spaces** are kinds of topological spaces. These conditions are examples of separation axioms.

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 T _{4}**: every two disjoint closed sets of

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.

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 for 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, a much larger class of spaces.

In topology, a **discrete space** is a particularly simple example of a topological space or similar structure, one in which the points form a *discontinuous sequence*, meaning they are *isolated* from each other in a certain sense. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, each singleton is an open set in the discrete topology.

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. Another name for general topology is **point-set 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 mathematics, the **Hilbert cube**, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube.

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**, 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 which 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.

In mathematics, the **particular point topology** is a topology where a set is open if it contains a particular point of the topological space. Formally, let *X* be any set and *p* ∈ *X*. The collection

In mathematics, a topological space is called **collectionwise normal** if for every discrete family *F*_{i} of closed subsets of there exists a pairwise disjoint family of open sets *U*_{i}, such that *F*_{i} ⊂ *U*_{i}. 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 demands that the above *U*_{i} are themselves a discrete family, which is stronger than pairwise disjoint.

- ↑ 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.

[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

- Willard, Stephen (1970).
*General topology*. Reading, Mass.: Addison-Wesley Pub. Co. p. 369. ISBN 9780486434797.

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