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.1
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}}} .2
A locally finite space is an Alexandrov space.3
A T1 space is locally finite if and only if it is discrete.4
References
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. 0-13-181629-2 ↩
Willard, Stephen (2016). "6". General topology. Mineola, N.Y: Dover Publications. ISBN 978-0-486-43479-7. 978-0-486-43479-7 ↩
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. 0-13-181629-2 ↩
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. https://doi.org/10.1155%2FS0161171201006482 ↩