The Lattice of Domains of an Extremally Disconnected Space 1
Zbigniew Karno · 1992
Let X be a topological space and let A be a subset of X . Recall that A is said to be a domain in X provided IntA ⊆ A ⊆ IntA (see [19], [7]). Recall also that A is said to be a(n) closed (open) domain in X if A = IntA (A = IntA, resp.) (see e.g. [9], [19]). It is well-known that for a given topological space all its closed domains form a Boolean lattice, and similarly all its open domains form a Boolean lattice, too (see e.g., [10], [2]). In [17] it is proved that all domains of a given topological space form a complemented lattice. One may ask whether the lattice of all domains is Boolean. The aim is to give an answer to this question. To present the main results we first recall the definition of a class of topological spaces which is important here. X is called extremally disconnected if for every open subset A of X the closure A is open in X [13] (comp. [6]). It is shown here, using Mizar System, that the lattice of all domains of a topological space X is modular iff X is extremally disconnected. Moreover, for every extremally disconnected space the lattice of all its domains coincides with both the lattice of all its closed domains and the lattice of all its open domains. From these facts it follows that the lattice of all domains of a topological space X is Boolean iff X is extremally disconnected. Note that we also review some of the standard facts on discrete, anti-discrete, almost discrete, extremally disconnected and hereditarily extremally disconnected topological spaces (comp. [9], [6]).