Finite topology is a mathematical concept which has several different meanings.
Finite topological space
A
finite topological space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements.
Finite topological spaces are often used to provide example ...
is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, the underlying set of which is finite.
In endomorphism rings
If ''A'' and ''B'' are
abelian groups
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 commut ...
then the finite topology on the group of homomorphisms Hom(''A'', ''B'') can be defined using the following
base of open neighbourhoods of zero.
:
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 a ...
s where we have ''A'' = ''B''. See section 14 of Krylov et al.
References
{{reflist
General topology