This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Topological invariants

### From Mathematics Is A Science

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:

**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.