Homeomorphisms

Definition: Homeomorphism

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces.

A homeomorphism between $X$ and $Y$ is a continuous bijection $f:X\to Y$ with a continuous inverse $f^{-1}:Y\to X$.

Definition: Homeomorphic Spaces

Two topological spaces $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ are homeomorphic if there exists a homeomorphism between them.

NOTATION

$$(X,{\tau }_X)\cong (Y,{\tau }_Y)$$

Theorem

The existence of a homeomorphism is an equivalence relation.

PROOF

TODO

Theorem: Composition of Homeomorphisms

Let $(X,{\tau }_X)$, $(Y,{\tau }_Y)$ and $(Z,{\tau }_Z)$ be topological spaces.

If $f:X\to Y$ and $g:Y\to Z$ are homeomorphisms, then their composition $g\circ f:X\to Z$ is also a homeomorphism.

PROOF

TODO

Local Homeomorphisms

Definition: Local Homeomorphism

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces.

A function $f:X\to Y$ is a local homeomorphism from $(X,{\tau }_X)$ to $(Y,{\tau }_Y)$ if each $x\in X$ has an open neighborhood $O$ with an open image $f(O)$ such that the restriction $f{|}_O$ is a homeomorphism between the subspaces $O$ and $f(O)$.

Definition: Locally Homeomorphic Spaces

A topological space $(X,{\tau }_X)$ is locally homeomorphic to another topological space $(Y,{\tau }_Y)$ if there exists a local homeomorphism from $(X,{\tau }_X)$ to $(Y,{\tau }_Y)$.

Theorem: Homeomorphism $⟹$ Local Homeomorphism

Every homeomorphism is also a local homeomorphism.

PROOF

TODO

Theorem

A local homeomorphism is a homeomorphism if and only if it is bijective.

PROOF

TODO