Topological invariants

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:

• compactness: ${\bf R} \neq [0,1] \neq (0,1)$;
• connectedness: point $\neq$ two points;
• simple connectedness: circle $\neq$ annulus $\neq$ sphere;
• manifold: square frame $\neq$ square frame with diagonal;
• manifold with boundary: circle $\neq [0,1]$;
• dimension (for manifolds): circle $\neq$ sphere;
• orientability (for manifolds): disk $\neq$ Mobius band, torus $\neq$ Klein bottle;
• homology: torus $\neq$ double torus.

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.