It can also be used to prove a metrization theorem of the same name, which applies to second-countable topological spaces.
The proof of this lemma utilizes the fact that the dyadic rationals are dense in the interval (0,1); it involves the construction of a function through an iteration scheme, sometimes called an onion function. It is used in the proof of the Tietze Extension Theorem, and its establishment of locales led to the formulation of the Tychnoff property. It can also be used to prove a metrization theorem of the same name, which applies to second-countable topological spaces. It states that for any two disjoint closed subsets of a normal topological space, there exists a continuous function from the space to the closed interval [0,1] which is 0 on one closed set and 1 on the other. FTP, name this fundamental lemma of classical topology.
The proof of this lemma utilizes the fact that the dyadic rationals are dense in the interval (0,1); it involves the construction of a function through an iteration scheme, sometimes called an onion function.
Urysohn's Lemma