Math and science::Topology

# Homeomorphisms. Homeomorphic spaces.

### Homeomorphisms and homeomorphic spaces.

Let $$X$$ and $$Y$$ be topological spaces.

Homeomorphism
A homeomorphism from $$X$$ to $$Y$$ is a [...].
Homeomorphic spaces
The spaces $$X$$ and $$Y$$ are homeomorphic iff [...].

$$X \cong Y$$ is the notation for $$X$$ to be homeomorphic to $$Y$$. 'Topologically equivalent' is an alternative term for 'homeomorphic'.

### When are two spaces homeomorphic?

To show that two spaces are homeomorphic, find a homeomorphism between them.

Roughly, two spaces are homeomorphic iff one can be deformed into the other by bending and reshaping, but without tearing or gluing.

### Equivalence relation

Being homeomorphic is an equivalence relation on the class of all topological spaces.

Reflexive
Let $$X$$ be a topological space. Then the identity map on $$X$$ is a homeomorphism.
Symmetric
Let $$f : X \to Y$$ be a homeomorphism. Then $$f^{-1} : Y \to X$$ is a homeomorphism.
Transitive
Let $$f : X \to Y$$ and $$g : Y \to Z$$ be homeomorphisms. Then $$g \circ f : X \to Z$$ is a homeomorphism.

Thus, $$X \cong X$$; $$X \cong Y \iff Y \cong X$$; $$X \cong Y \land Y \cong Z \implies X \cong Z$$.