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Finite topology

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Not to be confused with Cofinite 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 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.

U x 1 , x 2 , … , x n = { f ∈ Hom ⁡ ( A , B ) ∣ f ( x i ) = 0  for  i = 1 , 2 , … , n } {\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. Similarly, if R is a ring and M is a right R-module, then the finite topology on End R ( M ) {\displaystyle {\text{End}}_{R}(M)} is defined using the following system of neighborhoods of zero:

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

In vector spaces

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

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.

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.

Notes

  1. May 2001.
  2. Krylov, Mikhalev & Tuganbaev 2002, pp. 4598–4735.
  3. Abyazov & Maklakov 2023, p. 74.
  4. Kakutani & Klee 1963, pp. 55–58.
  5. Pazzis 2018, p. 2.
  6. Hoffman & Karcher 1995, p. 75.

References

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