Locally finite space

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 ${\displaystyle X}$ is metrizable without the countable basis requirement from Urysohn's theorem.^[1]

Definitions

Let ${\displaystyle T=(S,\tau )}$ be a topological space and let ${\displaystyle {\mathcal {F}}}$ be a set of subsets of ${\displaystyle S}$ Then ${\displaystyle {\mathcal {F}}}$ is locally finite if and only if each element of ${\displaystyle S}$ has a neighborhood which intersects a finite number of sets in ${\displaystyle {\mathcal {F}}}$.^[2]

A locally finite space is an Alexandrov space.^[1]

A T₁ space is locally finite if and only if it is discrete.^[3]