An abstract theory of topological subspaces

Extract

1. Introduction. The open subsets of a topological space X form a complete Brouwerian, i.e. distributive and pseudo-complemented lattice L(X). Many topological properties of X can be formulated as properties of L(X), although, in general, X is not determined by L(X). Obvious examples are quasi-compactness and connectivity: X is quasi-compact if any set of elements of L(X) whose join is I has a finite subset whose join is I; X is connected if no element of L(X) has a complement. These properties which we shall call ‘paratopological’, can be defined for lattices that are not of the form L(X), and many topological theorems can be proved in this more general context. The purpose of this paper is to develop sections of general topology as part of the theory of complete Brouwerian lattices (CBL).