Properties of topological properties
A property is: * Hereditary, if for every topological space and , the subspace has property P. * Weakly hereditary, if for every topological space and closed , the subspace has property P.Common topological properties
Cardinal functions
* The cardinality , ''X'', of the space ''X''. * The cardinality ''τ''(''X'') 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. * 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.) * 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. * 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. * 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'': ,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 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'': ,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 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 theCompactness
* Compact. A space isMetrizability
* Metrizable. A space is metrizable if it is homeomorphic to aMiscellaneous
* 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 isNon-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.See also
* Euler characteristic * Winding number * Characteristic class * Characteristic numbers * Chern class * Knot invariant * Linking number * Fixed-point property * Topological quantum number * Homotopy group and Cohomotopy group * Homology and cohomology * Quantum invariantCitations
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