Difference between revisions of "Compact-to-Hausdorff theorem"

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==Statement==
 
==Statement==
 
Given a [[Continuous map|continuous]] and [[Bijection|bijective]] function between two [[Topological space|topological spaces]] {{M|f:X\rightarrow Y}} where
 
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]]
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{{M|X}} is [[Compactness|compact]] and {{M|Y}} is [[Hausdorff space|Hausdorff]]
 
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* '''Then {{M|f}} is a [[Homeomorphism|homeomorphism]]'''<ref>Introduction to Topology - Nov 2013 - Lecture Notes - David Mond</ref>
 
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'''Then {{M|f}} is a [[Homeomorphism|homeomorphism]]'''<ref>Introduction to Topology - Nov 2013 - Lecture Notes - David Mond</ref>
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==Proof==
 
==Proof==
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{{Theorem Of|Topology}}
 
{{Theorem Of|Topology}}
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[[Category:Theorems involving compactness]]

Revision as of 05:06, 9 June 2015

Statement

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

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

References

  1. Jump up Introduction to Topology - Nov 2013 - Lecture Notes - David Mond