Compact-to-Hausdorff theorem

Statement

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

Then [ilmath]f[/ilmath] is a homeomorphism

Proof

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

Proof:

Let [ilmath]U\subseteq X[/ilmath] be a given open set

[ilmath]U[/ilmath] open [ilmath]\implies X-U[/ilmath] is closed [ilmath]\implies X-U[/ilmath] is compact

• (Using the compactness of [ilmath]X[/ilmath]) - a Closed set in compact space is compact)

[ilmath]\implies f(X-U)[/ilmath] is compact

• (Using Image of a compact set is compact)

[ilmath]\implies f(X-U)[/ilmath] is closed in [ilmath]Y[/ilmath]

• (Using Compact set in a Hausdorff space is closed

[ilmath]\implies Y-f(X-U)[/ilmath] is open in [ilmath]Y[/ilmath]

But [ilmath]Y-f(X-U)=f(U)[/ilmath]

• So we conclude [ilmath]f(U)[/ilmath] is open in [ilmath]Y[/ilmath]

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

References

Topology

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