Finite topology

A finite topological space is a topological space, the underlying set of which is finite.

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.

${\displaystyle U_{x_{1},x_{2},\ldots ,x_{n}}=\{f\in \operatorname {Hom} (A,B)\mid f(x_{i})=0{\mbox{ for }}i=1,2,\ldots ,n\}}$

This concept finds applications especially in the study of endomorphism rings where we have A = B. [1] Similarly, if R is a ring and M is a right R-module, then the finite topology on ${\displaystyle {\text{End}}_{R}(M)}$ is defined using the following system of neighborhoods of zero:[2]

${\displaystyle U_{X}=\{f\in {\text{End}}_{R}(M)\mid f(X)=0\}}$

In a vector space ${\displaystyle V}$, the finite open sets ${\displaystyle U\subset V}$ are defined as those sets whose intersections with all finite-dimensional subspaces ${\displaystyle F\subset V}$ are open. The finite topology on ${\displaystyle V}$ is defined by these open sets and is sometimes denoted ${\displaystyle \tau _{f}(V)}$. [3]

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.[4]

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.[5]

• Abyazov, A.N.; Maklakov, A.D. (2023), "Finite topologies and their properties in linear algebra", Russian Mathematics, 67 (1), doi:10.3103/s1066369x23010012, S2CID 256721835
• Hoffman, D.; Karcher, Hermann (1995), "Complete embedded minimal surfaces of finite total curvature", arXiv:math/9508213
• Kakutani, Shizuo; Klee, Victor (December 1963), "The finite topology of a linear space", Archiv der Mathematik, 14 (1): 55–58, doi:10.1007/bf01234921, S2CID 120704814
• Krylov, P.A.; Mikhalev, A.V.; Tuganbaev, A.A. (2002), "Properties of endomorphism rings of abelian groups I.", Journal of Mathematical Sciences, 112 (6): 4598–4735, doi:10.1023/A:1020582507609, MR 1946059, S2CID 120424104
• Pazzis, C. (2018), "On the finite topology of a vector space and the domination problem for a family of norms", arXiv:1801.09085 [math.GN]