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Topological invariants

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A topological invariant is a property of a topological space preserved under homeomorphisms. These properties allow one to tell whether two spaces are topologically equivalent. Sometimes. Other times, it doesn't.

Most examples of topological invariants are local: if $X$ and $Y$ are homeomorphic, then neighborhoods of points are homeomorphic too (see Locally homeomorphic spaces).

Examples:

For cell complexes, the last concept takes this form:

if the $|K|$ is homeomorphic to $|L|$ then $H(K)$ is isomorphic to $H(L)$.

Metatheorem. Every property expressed in terms of open sets only is a topological invariant.

Examples are all of the above. Non-examples are differentiability of manifolds, triangulation, etc.