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]] | + | {{M|X}} is [[Compactness|compact]] and {{M|Y}} is [[Hausdorff space|Hausdorff]] |
| − | + | * '''Then {{M|f}} is a [[Homeomorphism|homeomorphism]]'''<ref>Introduction to Topology - Nov 2013 - Lecture Notes - David Mond</ref> | |
| − | + | ||
| − | '''Then {{M|f}} is a [[Homeomorphism|homeomorphism]]'''<ref>Introduction to Topology - Nov 2013 - Lecture Notes - David Mond</ref> | + | |
==Proof== | ==Proof== | ||
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: Let {{M|U\subseteq X}} be a given open set | : 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]] | :: {{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]]) | + | ::* (Using the compactness of {{M|X}}) - a [[Closed set in a compact space is compact]]) |
:: {{M|\implies f(X-U)}} is compact | :: {{M|\implies f(X-U)}} is compact | ||
::* (Using [[Image of a compact set is compact]]) | ::* (Using [[Image of a compact set is compact]]) | ||
| Line 22: | Line 20: | ||
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''' | 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== | ||
<references/> | <references/> | ||
| − | {{Theorem|Topology}} | + | {{Theorem Of|Topology}} |
| + | [[Category:Theorems involving compactness]] | ||
Latest revision as of 12:36, 13 August 2015
Statement
Given a continuous and bijective function between two topological spaces f:X→Y where X is compact and Y is Hausdorff
- Then f is a homeomorphism[1]
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⊆X be a given open set
- U open ⟹X−U is closed ⟹X−U is compact
- (Using the compactness of X) - a Closed set in a compact space is compact)
- ⟹f(X−U) is compact
- ⟹f(X−U) is closed in Y
- ⟹Y−f(X−U) is open in Y
- But Y−f(X−U)=f(U)
- U open ⟹X−U is closed ⟹X−U is compact
- 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
- Jump up ↑ Introduction to Topology - Nov 2013 - Lecture Notes - David Mond
