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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a topological property or topological invariant is a property of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
that is invariant under
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
s. Alternatively, a topological property is a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map fo ...
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 set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
s. A common problem in topology is to decide whether two topological spaces are
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
or not. To prove that two spaces are ''not'' homeomorphic, it is sufficient to find a topological property which is not shared by them.

# Properties of topological properties

A property $P$ is: * Hereditary, if for every topological space $\left(X, \mathcal\right)$ and $X\text{'} \subset X$, the subspace $\left(X\text{'}, \mathcal, X\text{'}\right)$ has property P. * Weakly hereditary, if for every topological space $\left(X, \mathcal\right)$ and closed $X\text{'} \subset X$, the subspace $\left(X\text{'}, \mathcal, X\text{'}\right)$ has property P.

# Common topological properties

## Cardinal functions

* The
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
, ''X'', of the space ''X''. * The cardinality $\vert$''τ''(''X'')$\vert$ of the topology (the set of open subsets) 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''.

## 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 subsets, then there is a ''p'' such that the closure of 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 or Urysohn. A space is Urysohn if every two distinct points have disjoint ''closed'' neighbourhoods. T spaces are always T2. * Completely T2 or completely Hausdorff. A space is completely T2 if every two distinct points are
separated by a function In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
. 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 are
separated by a function In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
. * T, Tychonoff, Completely regular Hausdorff or Completely T3. A
Tychonoff space 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. A Tychonoff space refers to any completely regular space that is ...
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 In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0 ...
. * 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 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. T ...
if any two
separated sets In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
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 In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
. 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 Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...
if all of its points are completely isolated, i.e. if any subset is open. * Number of isolated points. The number of
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
s of a topological space.

## Countability conditions

* Separable. A space is separable (topology), separable if it has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
dense subset. * First-countable. A space is first-countable if every point has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
local base. * Second-countable. A space is second-countable if it has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
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'':  ,1nbsp;→ ''X'' with ''p''(0) = ''x'' and ''p''(1) = ''y''. Path-connected spaces are always connected. * Locally path-connected. A space is
locally path-connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...
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 In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
continuous map ''f'':  ,1nbsp;→ ''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 In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
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 In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within tha ...
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 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 consequ ...
.

## Compactness

* Compact. A space is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
if every
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
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 In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X. Every metric space is naturally a topological space, and for metric spaces, the notio ...
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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
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 In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
. 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 Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, si ...
if for every ''x'' and ''y'' in ''X'' there is a homeomorphism $f : X \to X$ such that $f\left(x\right) = y.$ 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 algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
s. * Reidemeister torsion * $\kappa$-resolvable. A space is said to be κ-resolvable (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 $\Delta\left(X\right)$-resolvable, where $\Delta\left(X\right) = \min\.$ Number $\Delta\left(X\right)$ 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 In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
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 $X = \R$ and $Y = \left(-\tfrac,\tfrac\right)$ be metric spaces with the standard metric. Then, $X \cong Y$ via the homeomorphism $\operatorname\colon X \to Y$. However, $X$ is complete but not bounded, while $Y$ is bounded but not complete.

*
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
*
Winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tur ...
* Characteristic class * Characteristic numbers * Chern class *
Knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...
* Linking number * Fixed-point property *
Topological quantum number In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers ar ...
*
Homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homoto ...
and Cohomotopy group * Homology and
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
* Quantum invariant

# References

* * {{refend 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 Homeomorphisms ru:Топологический инвариант