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


 * $$U_X = \{f\in \text{End}_R(M) \mid f(X) = 0\}$$

In vector spaces
In a vector space $$V$$, the finite open sets $$U\subset V$$ are defined as those sets whose intersections with all finite-dimensional subspaces $$F\subset V$$ are open. The finite topology on $$V$$ is defined by these open sets and is sometimes denoted $$\tau_f(V)$$.

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.

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.