Structures of topological type

Abstract

In this paper we continue the study of structures of various types initiated by the author in the earlier paper “Structures of extensions” (Ref. Zh. Mat., 1974, 4A361). The present paper is devoted to the so-called structure of topological type. By a structure of topological type on the set X is meant a topological structure, defined on some set obtained from X, and possibly additional sets, by a totally ordered sequence of operations of unions of sets, products of sets, and passage to the set of subsets. We study certain structures of topological type: bitopological (Sec. 2) and settopological (Sec. 3). A bitopological structure on the set X is any topological structure β on the set X×X, and a bitopological space is a pair (X,β). This concept is a natural extension of the concept of a bitopological space as a set X on which there are given two topological structures τ1 and τ2-these structures define a structure β=τ1×τ2 on the set X×X. A settopological structure on the set X is any topological structureζ on the setρχ={A¦A⊂χ. There are given representations of piecewise-linear structures (Sec. 4) and smooth structures (Sec. 5) as settopological structures.