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

Finite topological space

A finite topological space is a topological space, the underlying set of which is finite.

In endomorphism rings and modules

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.^[1]

${\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. ^[2] Similarly, if R is a ring and M is a right R-module, then the finite topology on ${\displaystyle {\text{End))_{R}(M)}$ is defined using the following system of neighborhoods of zero:^[3]

${\displaystyle U_{X}=\{f\in {\text{End))_{R}(M)\mid f(X)=0\))$

In vector spaces

In a vector space ${\displaystyle V}$, the finite open sets ${\displaystyle U\subset V}$ are defined as those sets whose intersections with all finite-dimensional subspaces ${\displaystyle F\subset V}$ are open. The finite topology on ${\displaystyle V}$ is defined by these open sets and is sometimes denoted ${\displaystyle \tau _{f}(V)}$. ^[4]

When V has uncountable dimension, this topology is not locally convex nor does it make V as topological vector space, but when V has countable dimension it coincides with both the finest vector space topology on V and the finest locally convex topology on V.^[5]

In manifolds

A manifold M is sometimes said to have finite topology, or finite topological type, if it is homeomorphic to a compact Riemann surface from which a finite number of points have been removed.^[6]