Finite topology
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 can be defined using the following base of open neighbourhoods of zero.This concept finds applications especially in the study of endomorphism rings where we have A = B.
Similarly, if R is a ring and M is a right R-module, then the finite topology on is defined using the following system of neighborhoods of zero:
In vector spaces
In a vector space, the finite open sets are defined as those sets whose intersections with all finite-dimensional subspaces are open. The finite topology on is defined by these open sets and is sometimes denoted.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.