Finite topology

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. ==Notes==