Compact-to-Hausdorff theorem

Statement

Given a continuous and bijective function between two topological spaces $f:X\rightarrow Y$ where $X$ is compact and $Y$ is Hausdorff

• Then $f$ is a homeomorphism

Proof

We wish to show $(f^{-1})^{-1}(U)$ is open (where $U$ is open in $X$), that is that the inverse of $f$ is continuous.

Proof:

Let $U\subseteq X$ be a given open set

$U$ open $\implies X-U$ is closed $\implies X-U$ is compact

• (Using the compactness of $X$) - a Closed set in a compact space is compact)

$\implies f(X-U)$ is compact

• (Using Image of a compact set is compact)

$\implies f(X-U)$ is closed in $Y$

• (Using Compact set in a Hausdorff space is closed

$\implies Y-f(X-U)$ is open in $Y$

But $Y-f(X-U)=f(U)$

• So we conclude $f(U)$ is open in $Y$

As $f=(f^{-1})^{-1}$ we have shown that a continuous bijective function's inverse is continuous, thus $f$ is a homeomorphism