Finite 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.

[math]\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\} }[/math]

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 [math]\displaystyle{ \text{End}_R(M) }[/math] is defined using the following system of neighborhoods of zero:

[math]\displaystyle{ U_X = \{f\in \text{End}_R(M) \mid f(X) = 0\} }[/math]

In vector spaces

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

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.

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", arXiv:math/9508213
• "The finite topology of a linear space", Archiv der Mathematik 14 (1): 55–58, December 1963, doi:10.1007/bf01234921 
• 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 
• 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]