Difference between revisions of "Urysohn lemma"
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The '''Urysohn lemma''' states: | The '''Urysohn lemma''' states: | ||
<blockquote> | <blockquote> | ||
| − | If | + | If ''A'', ''B'' are two disjoint, closed subsets of a [[normal space]] ''X'', then there exist a [[continuous function]] ''f'' : ''X'' → [0, 1], such that ''f(A)'' = 0 and ''f(B)'' = 1. |
</blockquote> | </blockquote> | ||
| − | The converse of the Urysohn lemma also holds. | + | The converse of the Urysohn lemma also holds: if there is such a continuous function for any two disjoint closed sets ''A'' and ''B'' in ''X'', then ''X'' is normal. |
| − | [[ | + | |
| + | [[Category:Topology]] | ||
Latest revision as of 21:15, July 13, 2016
The Urysohn lemma states:
If A, B are two disjoint, closed subsets of a normal space X, then there exist a continuous function f : X → [0, 1], such that f(A) = 0 and f(B) = 1.
The converse of the Urysohn lemma also holds: if there is such a continuous function for any two disjoint closed sets A and B in X, then X is normal.