Urysohn's lemma

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This article gives the statement, and possibly proof, of a basic fact in topology.


Let X 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 A,B are disjoint closed subsets of X. Then, there exists a continuous function f:X \to [0,1] such that f(a) = 0 for all a \in A, and f(b) = 1 for all b \in B.

Note that the T1 assumption is not necessary, so Urysohn's lemma also holds for normal-minus-Hausdorff spaces, which is what many point set topologists are referring to when they use the term normal space.

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