Urysohn's lemma: Difference between revisions
(New page: {{basic fact}} ==Statement== Let <math>X</math> be a normal space (i.e., a topological space that is T1 and where disjoint closed subsets can be separated by disjoint op...) |
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Let <math>X</math> be a [[normal space]] (i.e., a topological space that is [[T1 space|T1]] and where disjoint closed subsets can be separated by disjoint open subsets). Suppose <math>A,B</math> are disjoint [[closed subset]]s 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>. | Let <math>X</math> be a [[normal space]] (i.e., a topological space that is [[T1 space|T1]] and where disjoint closed subsets can be separated by disjoint open subsets). Suppose <math>A,B</math> are disjoint [[closed subset]]s 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>. | ||
==Related facts== | |||
* [[Tietze extension theorem]] | |||
* [[Hahn-Dieudonne-Tong insertion theorem]] | |||
Revision as of 03:13, 9 December 2008
This article gives the statement, and possibly proof, of a basic fact in topology.
Statement
Let be a normal space (i.e., a topological space that is T1 and where disjoint closed subsets can be separated by disjoint open subsets). Suppose are disjoint closed subsets of . Then, there exists a continuous function such that for all , and for all .