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 $X$ is metrizable without the countable basis requirement from Urysohn's theorem.{{cite book |last1=Munkres |first1=James Raymond |title=Topology |date=2000 |publisher=Prentice Hall |location=Upper Saddle River (N. J.) |isbn=0-13-181629-2 |pages=155–157 |edition=2nd |url=https://math.ucr.edu/~res/math205B-2018/Munkres%20-%20Topology.pdf |access-date=24 March 2025 |language=en}}

Definitions

Let $T = ( S, \tau )$ be a topological space and let $\mathcal{F}$ be a set of subsets of $S$ Then $\mathcal{F}$ is locally finite if and only if each element of $S$ has a neighborhood which intersects a finite number of sets in $\mathcal{F}$ .{{cite book |last1=Willard |first1=Stephen |title=General topology |date=2016 |publisher=Dover Publications |location=Mineola, N.Y |isbn=978-0-486-43479-7 |language=en |chapter=6}}

A locally finite space is an Alexandrov space.

A T₁ space is locally finite if and only if it is discrete.{{Cite journal |first1=Fumie |last1=Nakaoka |first2=Nobuyuki |last2=Oda |year=2001 |title=Some applications of minimal open sets |journal=International Journal of Mathematics and Mathematical Sciences |volume=29 |issue=8 |pages=471–476 |doi=10.1155/S0161171201006482|doi-access=free }}

References

Category:General topology

Category:Properties of topological spaces