Accumulation points of nowhere dense sets in $H$-closed spaces

Abstract

In this paper a ten year old problem by Kulpa and Szymanski is settled by constructing an example of a minimal Hausdorff space without isolated points which has a point that is not the accumulation point of any nowhere dense subset of the minimal Hausdorff space. Also, a result by Kulpa and Szymanski is extended by showing that a regular point in an H-closed space without isolated points is the accumulation point of some nowhere dense subset. A decade ago, Kulpa and Szymafnski [KS] showed that each point in a compact Hausdorff space without isolated points is the accumulation point of a nowhere dense set and they asked if the result is true in the setting of minimal Hausdorff spaces without isolated points. They noted that the result is false for arbitrary H-closed spaces without isolated points. In this paper, we settle this question by giving an example of a minimal Hausdorff space without isolated points and a point which is not the accumulation point of any nowhere dense set. Also, we extend Kulpa and Szymanski's result by showing that a regular point in an H-closed space without isolated points is the accumulation point of some nowhere dense subset. First, a few preliminary results and definitions are needed. All spaces considered in this paper are assumed to be Hausdorff. A space X is H-closed if X is closed in every space in which X is a subspace. An open set U in a space X is said to be regular open if U = int(cl U). A space X is semiregular if { U c X: U is regular open} is a base for the topology of X and is minimal Hausdorff if there is no strictly coarser Hausdorff topology on the space. Katetov [K] has shown that a space is minimal Hausdorff iff it is H-closed and semiregular. For a space X, let tX = X U {q: qi is a free open ultrafilter on X}. For each open set U c X, let oU= U U { E It X\X: int(clV) c U for some Ve 9/}. THEOREM 1. Let X be a semiregular space. The following are true: (a) [B, F, S] The family {oU: U is regular open in X} is a base for a minimal Hausdorff extension ttX of X. (b) [PW] For a regular open subset U of X, cl,,XoU = oU U cl xU. D Received by the editors May 29, 1984. 1980 Mathematics Subject Classification. Primary 54D25.