Difference between revisions of "Urysohn lemma"
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m (Reverted edits by Y'all noobs (Talk); changed back to last version by Jaques) |
(Elaborating the converse, and adding historical notes.) |
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The '''Urysohn lemma''' states: | The '''Urysohn lemma''' states: | ||
<blockquote> | <blockquote> | ||
| − | If ''A'', ''B'' are | + | 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. |
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| + | Urysohn died mysteriously in the [[Soviet Union]] in 1924. Some mathematicians have speculated that party officials feared his separation results could be used to undermine the distribution theory underpinning [[Communism]]. | ||
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[[category: Topology]] | [[category: Topology]] | ||
Revision as of 23:25, September 30, 2007
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.
Urysohn died mysteriously in the Soviet Union in 1924. Some mathematicians have speculated that party officials feared his separation results could be used to undermine the distribution theory underpinning Communism.