On the lattice equivalence of topological spaces

Author:

Finch P. D.

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.

Publisher

Cambridge University Press (CUP)

Reference5 articles.

1. A remark on complete lattices represented by sets;V´clav;Casopis. Pest Mat,1962

2. Lattice-equivalence of topological spaces

3. On complete lattices represented by sets;Stelleckii;Uspeki Mat. Nauk,1957

4. Representations of complete lattices by sets;Büchi;Portugaliae Math,1952

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