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Finite topology

From Wikipedia, the free encyclopedia

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

Finite topological space

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

In endomorphism rings and modules

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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.[1]

This concept finds applications especially in the study of endomorphism rings where we have A = B. [2] Similarly, if R is a ring and M is a right R-module, then the finite topology on is defined using the following system of neighborhoods of zero:[3]

In vector spaces

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In a vector space , the finite open sets are defined as those sets whose intersections with all finite-dimensional subspaces are open. The finite topology on is defined by these open sets and is sometimes denoted . [4]

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.[5]

In manifolds

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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.[6]

Notes

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References

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  • 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
  • May, Warren (2001), "The use of the finite topology on endomorphism rings", Journal of Pure and Applied Algebra, 163 (1): 107–117, doi:10.1016/S0022-4049(00)00159-6, MR 1847379
  • 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]