On the lattice equivalence of topological spaces

Abstract

A topology on a set X is defined by specifying a family of its subsets which has the properties (i) arbitrary set intersections of members of belong to , (ii) finite set unions of members of belong to and (iii) the empty set □ and the set X each belong to . The members of are called the closed subsets of X. If X is any subset of X then denotes the closure of X, that is, the set intersection of all closed subsets which contain X, however when X = {x} contains one point only we will denote by . The pair (X, ) is called a topological space or, in what follows, a T-space. By a T-lattice we mean a complete distributive lattice of sets in which arbitrary g.l.b. means arbitrary set intersection, finite l.u.b. means finite set union and which contains the empty set □ It is well-known, for example Birkhoff [1], that if (X, ) is a T-space and the members of are partially ordered by set inclusion then is a T-lattice.

Quasihomeomorphisms and meet-semilattice equivalences of generalized topological spaces

“…The characterization of lattice equivalences between topological spaces which are induced by homeomorphisms was given by Finch [5]. Now we give a similar result for GTS's.…”

“…The concept of quasihomeomorphisms was introduced in algebraic geometry by Grothendieck and Dieudonné [6]. Many results and characterizations of these concepts were investigated in [4,5,9]. In this paper we study the properties of quasihomeomorphisms and meet-semilattice equivalences of generalized topological spaces.…”

Quasi-homeomorphisms and lattice-equivalences of topological spaces

“…Yip Kai-Wing [2] 2 A continuous function/: X ->• Y from a topological space X into a topological space Y is said to be quasi-homeomorphism if the following conditions are satisfied:…”

On rings of sets

“…Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely joinirreducible elements -G. N. Raney [7]; join prime, completely join prime, and supercompact elements -V. K. Balachandran [1], [2]; ./^-sub-irreducible elements -J. R. Biichi [5]; and lattice bisectors -P. D. Finch [6]. Also meetirreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & O. Frink [4].…”

“…It is interesting to note that the elementary methods used in representing distributive lattices carry over completely and yield all these results, although this is hardly obvious when one considers special elements of the lattice. I wish to express my gratitude to Professor P. D. Finch, whose paper [6] was the inspiration for this work.…”