# Finite topology

Finite topology is a mathematical concept which has several different meanings.

## Finite topological space

A Finite topological space is a topological space whose underlying set is finite.

## In endomorphism rings

If A and B are abelian groups then the finite topology on the group of homomorphisms Hom(A, B) can be defined using the following base of open neighbourhoods of zero.

${\displaystyle U_{x_{1},x_{2},\ldots ,x_{n}}=\{f\in \operatorname {Hom} (A,B)\mid f(x_{i})=0{\mbox{ for }}i=1,2,\ldots ,n\}}$

This concept finds applications especially in the study of endomorphism rings where we have A = B. See section 14 of Krylov et al. [1]

## References

1. ^ Krylov, P.A.; Mikhalev, A.V.; Tuganbaev, A.A. (2002), "Properties of endomorphism rings of abelian groups I.", J. Math. Sci. (New York), 112: 4598–4735, MR 1946059