In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is, a neighborhood consisting of finitely many elements.

Background

The conditions for local finiteness were created by Jun-iti Nagata and Yury Smirnov while searching for a stronger version of the Urysohn metrization theorem. The motivation behind local finiteness was to formulate a new way to determine if a topological space is metrizable without the countable basis requirement from Urysohn's theorem.

Definitions

is locally finite if and only if has a neighborhood.

A locally finite space is an Alexandrov space.

A T1 space is locally finite if and only if it is discrete.

References