Finite topology
From Wikipedia, the free encyclopedia
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
Summarize
Perspective
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
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
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
References
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