View source for Compact-to-Hausdorff theorem

==Statement==
Given a [[Continuous map|continuous]] and [[Bijection|bijective]] function between two [[Topological space|topological spaces]] {{M|f:X\rightarrow Y}} where
{{M|X}} is [[Compactness|compact]] and {{M|Y}} is [[Hausdorff]]


'''Then {{M|f}} is a [[Homeomorphism|homeomorphism]]'''<ref>Introduction to Topology - Nov 2013 - Lecture Notes - David Mond</ref>

==Proof==
We wish to show {{M|(f^{-1})^{-1}(U)}} is [[Open set|open]] (where {{M|U}} is open in {{M|X}}), that is that the inverse of {{M|f}} is continuous.

'''Proof:'''
: Let {{M|U\subseteq X}} be a given open set
:: {{M|U}} open {{M|\implies X-U}} is closed {{M|\implies X-U}} is [[Compactness|compact]]
::* (Using the compactness of {{M|X}}) - a [[Closed set in compact space is compact]])
:: {{M|\implies f(X-U)}} is compact
::* (Using [[Image of a compact set is compact]])
:: {{M|\implies f(X-U)}} is closed in {{M|Y}}
::* (Using [[Compact set in a Hausdorff space is closed]]
:: {{M|\implies Y-f(X-U)}} is open in {{M|Y}}
:: But {{M|1=Y-f(X-U)=f(U)}}
* So we conclude {{M|f(U)}} is open in {{M|Y}}

As {{M|1=f=(f^{-1})^{-1} }} we have shown that a continuous bijective function's inverse is continuous, '''thus {{M|f}} is a homeomorphism'''
==References==
<references/>

{{Theorem|Topology}}