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 is metrizable without the countable basis requirement from Urysohn's theorem.[1]
Definitions
Let be a topological space and let be a set of subsets of Then is locally finite if and only if each element of has a neighborhood which intersects a finite number of sets in .[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
- ↑ 1.0 1.1 Munkres, James Raymond (2000) (in en). Topology (2nd ed.). Upper Saddle River (N. J.): Prentice Hall. pp. 155–157. ISBN 0-13-181629-2. https://math.ucr.edu/~res/math205B-2018/Munkres%20-%20Topology.pdf. Retrieved 24 March 2025.
- ↑ Willard, Stephen (2016). "6" (in en). General topology. Mineola, N.Y: Dover Publications. ISBN 978-0-486-43479-7.
- ↑ 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.
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