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Elaborating the converse, and adding historical notes.
The '''Urysohn lemma''' states:
The '''Urysohn lemma''' states:
<blockquote>
<blockquote>
If ''A'', ''B'' are 2 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.
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.
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]].
[[category: Topology]]
[[category: Topology]]