Urysohn

Urysohn is a Russian surname most often associated with Pavel Sergeyevich Urysohn (1888–1924), a mathematician who made foundational contributions to topology and metric geometry. His work helped establish several core ideas in early 20th‑century topology and provided tools used throughout analysis and geometry.

Urysohn's lemma states that in any normal topological space, for any two disjoint closed sets, there exists a continuous function from the space to the closed interval [0,1] that takes the value 0 on one set and 1 on the other. This result is fundamental for embedding spaces into cubes and for constructing partitions of unity.

Urysohn's metrization theorem gives conditions under which a topological space can be given a metric. In its common form, every regular space with a countable base is metrizable, a criterion that helped identify which spaces can be studied with metric methods.

One of Urysohn's lasting legacies is the Urysohn universal metric space, a separable and complete metric space that contains an isometric copy of every separable metric space. It is unique up to isometry and is homogeneous, meaning any isometry between finite subspaces extends to an isometry of the whole space. The Urysohn space serves as a universal object in metric geometry and has influenced model-theoretic approaches to metric structures.