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]

References

1. Munkres, James Raymond (2000). Topology (PDF) (2nd ed.). Upper Saddle River (N. J.): Prentice Hall. pp. 155–157. ISBN   0-13-181629-2 . Retrieved 24 March 2025.
2. Willard, Stephen (2016). "6". General topology. Mineola, N.Y: Dover Publications. ISBN   978-0-486-43479-7.
3. Nakaoka, Fumie; Oda, Nobuyuki (2001). "Some applications of minimal open sets". International Journal of Mathematics and Mathematical Sciences. 29 (8): 471–476. doi: 10.1155/S0161171201006482 .