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.
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.
Properties of topological properties
A property is:
Hereditary, if for every topological space and subset the
subspace has property
Weakly hereditary, if for every topological space and closed subset the subspace has property
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.
T2½ or Urysohn. A space is
Urysohn if every two distinct points have disjoint closed neighbourhoods. T2½ 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.)
T3½, 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.
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.
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.
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 with and . Arc-connected spaces are path-connected.
Simply connected. A space X is
simply connected if it is path-connected and every continuous map 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 is said to be metrizable if there exists a metric for such that the metric topology is identical with the topology
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 such that 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.
-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.
Non-topological properties
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.
[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
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.
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.
Properties of topological properties
A property is:
Hereditary, if for every topological space and subset the
subspace has property
Weakly hereditary, if for every topological space and closed subset the subspace has property
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.
T2½ or Urysohn. A space is
Urysohn if every two distinct points have disjoint closed neighbourhoods. T2½ 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.)
T3½, 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.
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.
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.
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 with and . Arc-connected spaces are path-connected.
Simply connected. A space X is
simply connected if it is path-connected and every continuous map 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 is said to be metrizable if there exists a metric for such that the metric topology is identical with the topology
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 such that 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.
-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.
Non-topological properties
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.
[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