Compact-to-Hausdorff theorem
From Maths
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^{[1]}
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 a compact space is compact)
- [ilmath]\implies f(X-U)[/ilmath] is compact
- [ilmath]\implies f(X-U)[/ilmath] is closed in [ilmath]Y[/ilmath]
- [ilmath]\implies Y-f(X-U)[/ilmath] is open in [ilmath]Y[/ilmath]
- But [ilmath]Y-f(X-U)=f(U)[/ilmath]
- [ilmath]U[/ilmath] open [ilmath]\implies X-U[/ilmath] is closed [ilmath]\implies X-U[/ilmath] is compact
- 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
- ↑ Introduction to Topology - Nov 2013 - Lecture Notes - David Mond