On Some Categories of Sequential Spaces

The sequential spaces dealt with in the paper have arbitrary ordinals as domains of sequences and are subjected, besides the axiom of convergence of constant sequences, to only one other, quite special axiom. Categories of these sequential spaces are studied and the results are then applied to categories of associated closure spaces. It is a usual way to deene a topological structure on a set X by introducing a convergence on X, i.e. a map which assigns to any point x 2 X a set of sequences (of points of X) converging to x and which is subjected to certain axioms. The topological structure on X is then given by a closure operation where the closure of a subset A X is the set of all limits of sequences contained in A. The most known topological structures obtained by this way are the Fr echet S-spaces-see e.g. 4]. Another kind of such topological structures is studied in this paper. But here, in contrast to the classical access, domains of sequences are considered to be arbitrary ordinals (and not only the least innnite ordinal). Also the axiom prescribed besides the usual axiom of convergence of constant sequences is quite special. We show that closure spaces induced by these sequential spaces have a useful categorical property: they form an exponential supercategory of the category of nitely generated topological spaces. Let X be a set, > 0 an ordinal, and : X ! exp(X) a map such that (i) (x i ji <) 2 (x) whenever x i = x for all ordinals i < , (ii) (x i ji <) 2 (x) , for each ordinal j; 0 < j < , there is a sequence (y j i ji <) 2 (x j) with fy j i ; i < g fx i ; i < jg, and there is a sequence (y i ji <) 2 (x) with fy i ; i < g fx i ; i < g. The pair (X;) is called a sequential space of type. Given a pair of sequential spaces of type , (X;) and (Y;), by a homomorphism of (X;) into (Y;) we understand a map f : X ! Y with (x i ji <) 2 (x)) (f(x i)ji <) 2 (f(x)). We denote by Seq the ((nitely productive) category of sequential spaces of type with homomorphisms as morphisms. (All categories are …