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 X {\displaystyle X} is metrizable without the countable basis requirement from Urysohn's theorem.
Definitions Let T = ( S , τ ) {\displaystyle T=(S,\tau )} be a topological space and let F {\displaystyle {\mathcal {F}}} be a set of subsets of S {\displaystyle S} Then F {\displaystyle {\mathcal {F}}} is locally finite if and only if each element of S {\displaystyle S} has a neighborhood which intersects a finite number of sets in F {\displaystyle {\mathcal {F}}} . A locally finite space is an Alexandrov space. A T1 space is locally finite if and only if it is discrete.
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