Finite topology

Last updated October 07, 2023

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 and modules

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.^{[ citation needed ]}

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 ${\text{End}}_{R}(M)$ is defined using the following system of neighborhoods of zero^[2]:

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

In vector spaces

In a vector space $V$ , 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 $\tau _{f}(V)$ . ^[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]

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.^[5]

Notes

↑ Krylov 2002, p.4598–4735
↑ Abyazov and Maklakov 2023, p.74
↑ Kakutani and Klee 1963, p.55-58
↑ Pazzis 2018, p.2
↑ Hoffman and Karcher 1995, p.75

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References

Abyazov, A.N.; Maklakov, A.D. (2023), "Finite topologies and their properties in linear algebra", Russian Mathematics, 67 (1), doi:10.3103/s1066369x23010012
Hoffman, D.; Karcher, Hermann (1995), Complete embedded minimal surfaces of finite total curvature, doi:10.48550/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
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
Pazzis, C. (2018), On the finite topology of a vector space and the domination problem for a family of norms, doi:10.48550/arXiv.1801.09085

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[1] Krylov 2002, p.4598–4735

[2] Abyazov and Maklakov 2023, p.74

[3] Kakutani and Klee 1963, p.55-58

[4] Pazzis 2018, p.2

[5] Hoffman and Karcher 1995, p.75

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