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.

$U_{x_1,x_2,\ldots,x_n}=\{f\in\operatorname{Hom}(A,B)\mid f(x_i)=0 \mbox{ for } i=1,2,\ldots,n\}$

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