On connected irresolvable Hausdorff spaces

Abstract

Hewitt [2, p. 327] has raised, and Padmavally [4] has answered to the affirmative, the question of the existence of connected irresolvable Hausdorff spaces. The purpose of this note is to prove a general existence theorem for connected irresolvable Hausdorff spaces which shows that the class of such spaces is more numerous than previously supposed. In the discussion preceding Theorem 2 the underlying point set of a topological space will be denoted by X. A topology on X will be denoted by either R or T. When it is necessary to distinguish between different topologies on the same set X, subscripts will be used. We recall some definitions and notations. If R1 and R2 are two topologies for X, R2 is an expansion of R, if R,CR2. A topology R for X is irresolvable if in the space (X, R) there is no dense set D for which X -D is also dense. The dispersion character of a topology R for X, denoted by A(R), is the least cardinality of a nonempty open set in R. Throughout this paper we shall be concerned with a special class of expansions of a given topology on X. We make the following DEFINITION. Let (X, R) be a topological space, {Da: -aCA } be the set of all R-dense subsets of X. An expansion R* of R is admissible if R* has a subbasis of the form RU {D:: f3z-BCA } and if each of the sets D: is R*-dense. We note that any admissible expansion R* of R has a unique subbasis S*=RUJ{D#:GBCA} with B maximal. S*=RU{D:D is R*-dense and R*-open } is such a subbasis for R*. This subbasis will be called the admissible subbasis for R* and the subset BCA will be called the index class corresponding to R*.