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.

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.

Common topological properties

Cardinal functions

Main article: Cardinal function § Cardinal functions in topology

Separation

Main article: Separation axiom

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

Countability conditions

See also: Axiom of countability

Connectedness

Compactness

Metrizability

Miscellaneous

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.

See also

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Citations

  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.

References

[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