Compact to Hausdorff implies closed - Revision history

Vipul at 06:12, 23 December 2009

2009-12-23T06:12:08Z

← Older revision		Revision as of 06:12, 23 December 2009
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	* [[Surjection from compact to Hausdorff implies quotient\|Any surjective continuous map from a compact space to a Hausdorff space is a quotient map]]		* [[Surjection from compact to Hausdorff implies quotient\|Any surjective continuous map from a compact space to a Hausdorff space is a quotient map]]
	* [[Injection from compact to Hausdorff implies embedding\|Any continuous injective map from a compact space to a Hausdorff space is a subspace embedding]]		* [[Injection from compact to Hausdorff implies embedding\|Any continuous injective map from a compact space to a Hausdorff space is a subspace embedding]]

			==Facts used==

			# [[uses::Compactness is weakly hereditary]]: Any closed subset of a compact space is compact in the subspace topology.
			# [[uses::Compactness is continuous image-closed]]: The image of a compact space under a continuous map is a compact space.
			# [[uses::Hausdorff implies KC]]: Any compact subset of a Hausdorff space is closed.

	==Proof==		==Proof==

	~~We need~~ to ~~show that given~~ any closed subset of ~~the compact space~~, ~~its image~~ is closed ~~in the Hausdorff space~~.		'''Given''': A compact space <math>X</math>, a Hausdorff space <math>Y</math>, a continuous map <math>f:X \to Y</math>.

			'''To prove''': For any closed subset <math>A</math> of <math>X</math>, <math>f(A)</math> is a closed subset of <math>Y</math>.

	~~We reason in the following steps~~:		'''Proof''':

	* The closed subset of the compact space, is ~~itself~~ compact in the [[subspace topology~~]].~~ This is ~~because any closed subset of a~~ compact ~~space~~ is compact~~. {{proofat~~\|~~[[Compactness~~ is ~~weakly hereditary]]}}~~		# <math>A</math> is compact under the subspace topology: This follows from the given datum that <math>X</math> is compact and fact (1).
	* Thus, the ~~image~~ of ~~this subset,~~ in the ~~target space~~, is ~~also~~ compact in the [[subspace topology~~]]. This follows~~ from ~~the fact that an image of a compact space under a continuous map is also compact~~. ~~{{further\|[[Compactness is continuous image-closed]]}}~~		# <math>f(A)</math> is compact under the subspace topology in <math>Y</math>: First, note that the map <math>f\|_A: A \to Y</math> is continuous, because it is the composite of the inclusion of <math>A</math> in <math>X</math> with the map <math>f</math>, both of which are continuous. Thus, by fact (2), <math>f(A)</math> is compact with the subspace topology from <math>Y</math>.
	* The image of this subset is a compact subset of a Hausdorff space, hence is closed in ~~the Hausdorff space.~~ This follows from the ~~fact~~ that ~~any compact subset of a~~ Hausdorff ~~space is closed~~. ~~{{further\|[[Hausdorff implies KC]]}}~~		# <math>f(A)</math> is closed in <math>Y</math>: This follows from the previous step, the given datum that <math>Y</math> is Hausdorff, and fact (3).

Vipul: 2 revisions

2008-05-11T19:41:52Z

2 revisions

← Older revision	Revision as of 19:41, 11 May 2008
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Vipul at 17:18, 13 January 2008

2008-01-13T17:18:10Z

← Older revision		Revision as of 17:18, 13 January 2008
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	==Statement==		==Statement==

	Any [[continuous map]] from a [[compact space]] to a [Hausdorff space]] is a [[closed map]] i.e. the image of any closed set is closed.		Any [[continuous map]] from a [[compact space]] to a [[Hausdorff space]] is a [[closed map]] i.e. the image of any closed set is closed.

	==Applications==		==Applications==

Vipul: New page: ==Statement== Any continuous map from a compact space to a [Hausdorff space]] is a closed map i.e. the image of any closed set is closed. ==Applications== * [[Surjection fro...

2008-01-13T17:17:54Z

New page: ==Statement== Any continuous map from a compact space to a [Hausdorff space]] is a closed map i.e. the image of any closed set is closed. ==Applications== * [[Surjection fro...

New page

==Statement==

Any [[continuous map]] from a [[compact space]] to a [Hausdorff space]] is a [[closed map]] i.e. the image of any closed set is closed.

==Applications==

* [[Surjection from compact to Hausdorff implies quotient|Any surjective continuous map from a compact space to a Hausdorff space is a quotient map]]
* [[Injection from compact to Hausdorff implies embedding|Any continuous injective map from a compact space to a Hausdorff space is a subspace embedding]]

==Proof==

We need to show that given any closed subset of the compact space, its image is closed in the Hausdorff space.

We reason in the following steps:

* The closed subset of the compact space, is itself compact in the [[subspace topology]]. This is because any closed subset of a compact space is compact. {{proofat|[[Compactness is weakly hereditary]]}}
* Thus, the image of this subset, in the target space, is also compact in the [[subspace topology]]. This follows from the fact that an image of a compact space under a continuous map is also compact. {{further|[[Compactness is continuous image-closed]]}}
* The image of this subset is a compact subset of a Hausdorff space, hence is closed in the Hausdorff space. This follows from the fact that any compact subset of a Hausdorff space is closed. {{further|[[Hausdorff implies KC]]}}