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 P is:

Common topological properties

Cardinal functions

Separation

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

Countability conditions

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 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 be metric spaces with the standard metric. Then, X \cong Y via the homeomorphism. However, X is complete but not bounded, while Y is bounded but not complete.

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