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

Finite topological space

finite topological space In mathematics, a finite topological space is a topological space for which the underlying set (mathematics), point set is finite set, finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are ...

is a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

, the underlying set of which is finite.

In endomorphism rings and modules

If ''A'' and ''B'' are

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

s 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_=\

This concept finds applications especially in the study of

endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...

s where we have ''A'' = ''B''. Similarly, if ''R'' is a ring and ''M'' is a right ''R''- module, then the finite topology on

\text_R(M)

is defined using the following system of neighborhoods of zero: :

U_X = \

In vector spaces

In a

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

, 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

References

* * * * * * {{cite arXiv, last1=Pazzis , first1=C. , title=On the finite topology of a vector space and the domination problem for a family of norms , eprint=1801.09085 , year=2018 , class=math.GN , mode=cs2 General topology