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]
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Hoffman, D.; Karcher, Hermann (1995), "Complete embedded minimal surfaces of finite total curvature",
arXiv:math/9508213
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,
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10.1023/A:1020582507609,
MR1946059,
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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,
MR1847379
Pazzis, C. (2018), "On the finite topology of a vector space and the domination problem for a family of norms",
arXiv:1801.09085 [
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