The lemma, sometimes called "the first nontrivial fact of point set topology", is often used to construct continuous functions with various properties; 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.
Note that in the statement above, we do not, and in general cannot, require that f(x) ≠ 0 and ≠ 1 for x outside of A and B. This is only possible in perfectly normal spaces.
Proof sketch For every dyadic fraction r ∈ (0,1), we are going to construct an open subset U(r) of X such that:
In order to construct the sets U(r), we actually do a little bit more: we construct sets U(r) and V(r) such that
This construction proceeds by mathematical induction. Since X is normal, we can find two disjoint open sets U(1/2) and V(1/2) which contain A and B, respectively. Now assume that n≥1 and the sets U(a/2^{n}) and V(a/2^{n}) have already been constructed for a = 1,...,2^{n}1. Since X is normal, we can find two disjoint open sets which contain the complement of V(a/2^{n}) and the complement of U((a+1)/2^{n}), respectively. Call these two open sets U((2a+1)/2^{n+1}) and V((2a+1)/2^{n+1}), and verify the above three conditions.
The Mizar project has completely formalized and automatically checked a proof of Urysohn's lemma in the URYSOHN3 file (http://www.mizar.org/JFM/Vol13/urysohn3).
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