The Lattice of Domains of a Topological Space 1

Let T be a topological space and let A be a subset of T . Recall that A is said to be a closed domain of T if A = IntA and A is said to be an open domain of T if A = IntA (see e.g. [8], [14]). Some simple generalization of these notions is the following one. A is said to be a domain of T provided IntA ⊆ A ⊆ IntA (see [14] and compare [7]). In this paper certain connections between these concepts are introduced and studied. Our main results are concerned with the following well–known theorems (see e.g. [9], [1]). For a given topological space all its closed domains form a Boolean lattice, and similarly all its open domains form a Boolean lattice, too. It is proved that all domains of a given topological space form a complemented lattice. Moreover, it is shown that both the lattice of open domains and the lattice of closed domains are sublattices of the lattice of all domains. In the beginning some useful theorems about subsets of topological spaces are proved and certain properties of domains, closed domains and open domains are discussed.