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

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. See section 14 of Krylov et al. [1]

References

Krylov, P.A.; Mikhalev, A.V.; Tuganbaev, A.A. (2002), "Properties of endomorphism rings of abelian groups I.", J. Math. Sci. (New York), 112 (6): 4598–4735, doi:10.1023/A:1020582507609, MR 1946059, S2CID 120424104

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[1] Krylov, P.A.; Mikhalev, A.V.; Tuganbaev, A.A. (2002), "Properties of endomorphism rings of abelian groups I.", J. Math. Sci. (New York), 112 (6): 4598–4735, doi:10.1023/A:1020582507609, MR 1946059, S2CID 120424104