Normal Hausdorff implies functionally Hausdorff - Revision history

Vipul: moved Normal implies Urysohn to Normal Hausdorff implies functionally Hausdorff

2012-01-27T22:15:32Z

moved Normal implies Urysohn to Normal Hausdorff implies functionally Hausdorff

← Older revision	Revision as of 22:15, 27 January 2012
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Vipul: /* Facts used */

2009-10-26T19:36:50Z

Facts used

← Older revision		Revision as of 19:36, 26 October 2009
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	==Facts used==		==Facts used==

	# [[uses::Urysohn's lemma]]: If <math>X</math> is a normal space and <math>A,B</math> are disjoint closed subsets of <math>X</math>, then there exists a continuous function <math>f:X \to [0,1]</math> such that <math>f(a) = 0</math> for all <math>a \in A</math> and <math>f(b) = 1</math> for all math>b \in B</math>.		# [[uses::Urysohn's lemma]]: If <math>X</math> is a normal space and <math>A,B</math> are disjoint closed subsets of <math>X</math>, then there exists a continuous function <math>f:X \to [0,1]</math> such that <math>f(a) = 0</math> for all <math>a \in A</math> and <math>f(b) = 1</math> for all <math>b \in B</math>.

	==Proof==		==Proof==

Vipul: Created page with '{{topospace property implication| stronger = normal space| weaker = Urysohn space}} ==Statement== Any normal space is a Urysohn space. ==Definitions used== ===Normal …'

2009-10-26T19:36:20Z

Created page with '{{topospace property implication| stronger = normal space| weaker = Urysohn space}} ==Statement== Any normal space is a Urysohn space. ==Definitions used== ===Normal …'

New page

{{topospace property implication|
stronger = normal space|
weaker = Urysohn space}}

==Statement==

Any [[normal space]] is a [[Urysohn space]].

==Definitions used==

===Normal space===

{{further|[[Normal space]]}}

A normal space is a topological space <math>X</math> that is a [[T1 space]] and satisfies the following: for any two disjoint closed subsets <math>A,B</math> of <math>X</math>, there are disjoint open subsets <math>U,V</math> of <math>X</math> containing <math>A,B</math> respectively.

===Urysohn space===

{{further|[[Urysohn space]]}}

A Urysohn space is a topological space <math>X</math> such that, for any two distinct points <math>x,y \in X</math>, there is a continuous function <math>f:X \to [0,1]</math> such that <math>f(x) = 0</math> and <math>f(y) = 1</math>.

==Facts used==

# [[uses::Urysohn's lemma]]: If <math>X</math> is a normal space and <math>A,B</math> are disjoint closed subsets of <math>X</math>, then there exists a continuous function <math>f:X \to [0,1]</math> such that <math>f(a) = 0</math> for all <math>a \in A</math> and <math>f(b) = 1</math> for all math>b \in B</math>.

==Proof==

'''Given''': A topological space <math>X</math> that is normal.

'''To prove''': <math>X</math> is a Urysohn space.

'''Proof''': Suppose <math>x,y</math> are distinct points of <math>X</math>. Since <math>X</math> is normal, it is <math>T_1</math>, and <math>\{ x \} ,\{ y \}</math> are both closed points. Fact (1) then provides the function <math>f:X \to [0,1]</math> with <math>f(x) = 0</math> and <math>f(y) = 1</math>.