Filter convergence via sequential convergence. (English) Zbl 0591.54003

Beattie, R.; Butzmann, Heinz-Peter; Herrlich, Horst

Commentat. Math. Univ. Carol. 27, 69-81 (1986).

Filter convergence structures as used in this paper are of the type introduced as convergence functions by D. C. Kent [Fundam. Math. 54, 125-133 (1964; Zbl 0122.413)] as a generalization of the convergence structure (Limesräume) of H. R. Fischer [Math. Ann. 137, 269-303 (1959; Zbl 0086.088)]. As stated in the authors’ abstract, the goal is to examine the forgetful functor between categories of filter and sequential convergence spaces. Said functor is shown to have a left adjoint. The restriction to the category of all first countable filter convergence spaces has both a left and a right adjoint and a suitable domain-codomain restriction is topological, where the second restriction is to the subcategories of first countable filter convergence structures and sequential spaces spaces satisfying condition (FL) of R. Frič and D. C. Kent [Abh. Akad. Wiss. DDR, Abt. Math. Naturwiss. Tech. 1979, Nr. 4N, 33-36 (1980; Zbl 0457.54002)]. Several equivalences of condition (FL) are also given.

Reviewer: C.V.Riecke

MSC:

54A20	Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
18A40	Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
54B30	Categorical methods in general topology
18B30	Categories of topological spaces and continuous mappings (MSC2010)
54D55	Sequential spaces

Keywords:

topological functor; sequential convergence spaces; left adjoint; category of all first countable filter convergence spaces; right adjoint

Citations:

Zbl 0122.413; Zbl 0086.088; Zbl 0457.54002