Finite topology

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

Finite topological space

A Finite topological space is a topological space whose underlying set 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]

gollark: They may just have never cared enough to study it.

gollark: Unless you want to do really heavy stuff you could probably get away with just *one* Pi and an external SSD.

gollark: Lots of programs seem to require the horribleness of cmake or those weird `configure` scripts.

gollark: If you don't you'll probably die of random chance eventually anywya.

gollark: Yes, just save system state often, have offsite backups of things which will turn on if the main one goes down, and have batteries.

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: 4598–4735, MR 1946059

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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: 4598–4735, MR 1946059