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 $f^{-1}:Y\to X$.

Criteria for Homeomorphisms

Theorem: Equivalent Definition

A bijection $f:X\to Y$ between two topological spaces $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ is a homeomorphism if and only if the image of each open subset of $(X,{\tau }_X)$ is open in $(Y,{\tau }_Y)$ and the inverse image of each open subset in $(Y,{\tau }_Y)$ is open in $(X,{\tau }_X)$.

PROOF

We have to prove two things:

• (I) If $f:X\to Y$ is a homeomorphism, then the image $f(U)$ of each open subset $U$ of $(X,{\tau }_X)$ is open in $(Y,{\tau }_Y)$.
• (II) If $f:X\to Y$ is a bijection and the image $f(U)$ of each open subset $U$ of $(X,{\tau }_X)$ is open in $(Y,{\tau }_Y)$, then $f$ is a homeomorphism.

Proof of (I):

Proof of (II):

TODO

Theorem: Homeomorphism via Open Maps

A bijection $f:X\to Y$ between two topological spaces $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ is a homeomorphism if and only if it is an open map.

PROOF

TODO

Theorem: Homeomorphism via Closed Maps

A bijection $f:X\to Y$ between two topological spaces $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ is a homeomorphism if and only if it is a closed map.

PROOF

TODO

Properties

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

Theorem: Homeomorphism $⟹$ Local Homeomorphism

Every homeomorphism is also a local homeomorphism.

PROOF

TODO