Urysohn's lemma

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In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function.

Urysohn's lemma is sometimes called "the first non-trivial fact of point set topology" and is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all metric spaces and all compact Hausdorff spaces are normal. The lemma is generalized by (and usually used in the proof of) the Tietze extension theorem.

The lemma is named after the mathematician Pavel Samuilovich Urysohn.


Formal statement

Two disjoint closed subsets A and B of a topological space X are said to be separated by neighbourhoods if there are neighbourhoods U of A and V of B that are also disjoint. A and B are said to be separated by a function if there exists a continuous function f from X into the unit interval [0,1] such that f(a) = 0 for all a in A and f(b) = 1 for all b in B. Any such function is called a Urysohn function for A and B.

A normal space is a topological space in which any two disjoint closed sets can be separated by neighbourhoods. Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a function.

The sets A and B need not be precisely separated by f, i.e., we do not, and in general cannot, require that f(x) ≠ 0 and ≠ 1 for x outside of A and B. This is possible only in perfectly normal spaces.

Urysohn's lemma has led to the formulation of other topological properties such as the 'Tychonoff property' and 'completely Hausdorff spaces'. For example, a corollary of the lemma is that normal T1 spaces are Tychonoff.

Sketch of proof

For every dyadic fraction r ∈ (0,1), we are going to construct an open subset U(r) of X such that:

Once we have these sets, we define f(x) = inf { r : xU(r) } for every xX. Using the fact that the dyadic rationals are dense, it is then not too hard to show that f is continuous and has the property f(A) ⊆ {0} and f(B) ⊆ {1}.

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