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Topological equivalence

Homeomorphisms, i.e. a function that is continuous along with its inverse, create an equivalence relation on the set of all topological spaces.

Examples, positive first..

Prove: interval ~ interval for open, closed, or half open.

For the open: $(a,b)\sim (0,1)$. Define $f:(0,1) \to (a,b)$ with $$f(x) = a(1-x)+bx.$$ Same for closed, and half-open. /

It makes sense to call two spaces topologically equivalent if they are homeomorphic. We use the following notation for that: $$X \sim Y.$$

Q: classify closed subsets of ${\bf R}^2$ homeomorphic to ${\bf S}^1$, the circle.

Q: A polygon/graph homeomorphic to ${\bf S}^1$.

This is not an entirely new situation. In linear algebra the equivalence relation between vector spaces is provided by isomorphisms, same in modern algebra. There are also classification theorems: two finitely dimensional vector spaces are isomorphic iff their dimensions are equal and two finitely generated abelian groups are isomorphic iff their ranks and torsions coincide. This means that these characteristics are complete invariants. Homology is a partial invariant:

Theorem. If two topological spaces are homeomorphic then their homologies are equal. More precisely, if $|K| \approx |L|$, where $K$ and $L$ are cubical complexes, then $H(K) = H(L)$.

In other words, homology is a topological invariant. However, the converse isn't true:

$H($point$) = H($closed cell$)$,

even though points and cells can't be homeomorphic.

The theorem is mostly used to prove that two spaces aren't homeomorphic, such as spheres of different dimensions:

${\bf S}^i \approx {\bf S}^j$ iff $i = j$.

Other characteristics are used in the same way: compactness, connectedness, separation axioms, etc.