Vipul at 04:37, 30 January 2014

2014-01-30T04:37:41Z

← Older revision		Revision as of 04:37, 30 January 2014
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			[[Difficulty level::1\| ]]
	==Statement==		==Statement==

Vipul at 16:21, 21 July 2008

2008-07-21T16:21:34Z

← Older revision		Revision as of 16:21, 21 July 2008
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	{{topospace metaproperty ~~satisaction~~}}		{{topospace metaproperty satisfaction}}

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Vipul: New page: {{topospace metaproperty satisaction}} ==Statement== ===Property-theoretic statement=== The property of topological spaces of being a Hausdorff space ===Statement with symbols==...

2008-07-21T16:20:14Z

New page: {{topospace metaproperty satisaction}} ==Statement== ===Property-theoretic statement=== The property of topological spaces of being a Hausdorff space ===Statement with symbols==...

New page

{{topospace metaproperty satisaction}}

==Statement==

===Property-theoretic statement===

The [[property of topological spaces]] of being a [[Hausdorff space]]
===Statement with symbols===

Suppose <math>(X,\tau)</math> is a topological space and <math>\tau'</math> is a [[finer topology]] on <math>X</math> than <math>\tau</math>.

==Proof==

'''Given''': A Hausdorff topological space <math>(X,\tau)</math>, and a topology <math>\tau'</math> on <math>X</math> that is [[finer topology|finer]] than <math>\tau</math>

'''To prove''': <math>(X,\tau')</math> is Hausdorff

'''Proof''': We need to show that for points <math>x \ne y</math> in <math>X</math>, there exist open sets <math>U,V</math> in the topology <math>\tau'</math> such that <math>x \in U, y \in V</math>, and <math>U \cap V</math> is empty.

Since <math>\tau</math> gives a Hausdorff topology, we can find open sets <math>U,V</math> in the topology <math>\tau</math>, such that <math>x \in U, y \in V</math> and <math>U \cap V</math> is empty. And since <math>\tau'</math> is finer than <math>\tau</math>, the sets <math>U,V</math> satisfy the condition in <math>\tau'</math> as well.

Hausdorffness is refining-preserved - Revision history

Vipul at 04:37, 30 January 2014

Vipul at 16:21, 21 July 2008

Vipul: New page: {{topospace metaproperty satisaction}} ==Statement== ===Property-theoretic statement=== The property of topological spaces of being a Hausdorff space ===Statement with symbols==...