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.Template:Sfn

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

This concept finds applications especially in the study of endomorphism rings where we have A = B. Template:Sfn 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:Template:Sfn

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

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 ${\tau }_f(V)$. Template:Sfn

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.Template:Sfn

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.Template:Sfn

References

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