H-closed and countably compact extensions

Countably I-compact Spaces

We introduce the class of countably I-compact spaces as a proper subclass of countably S-closed spaces. A topological space (X,T) is called countably I-compact if every countable cover of X by regular closed subsets contains a finite subfamily whose interiors cover X. It is shown that a space is countably I-compact if and only if it is extremally disconnected and countably S-closed. Other chara...

Topology Proceedings 5 (1980) pp. 223-229: TUNNELS, TIGHT GAPS, AND COUNTABLY COMPACT EXTENSIONS OF N

Brandt extensions and primitive topologically periodic inverse topological semigroups

In this paper we find sufficient conditions on primitive inverse topological semigroup S under which: the inversion inv : (H(S)) (H(S)) is continuous; we show that every topologically periodic countable compact primitive inverse topological semigroups with closed H-classes is topologically isomorphic to an orthogonal sum P i2= Bi (Gi) of topological Brandt extensions Bi (Gi) of countably compac...

PFA(S)[S] and countably compact spaces

We show a number of undecidable assertions concerning countably compact spaces hold under PFA(S)[S]. We also show the consistency without large cardinals of every locally compact, perfectly normal space is paracompact.

Countably compact hyperspaces and Frolík sums

Let H 0(X) (H(X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H(X) is to characterize those X for which H(X) is countably compact. We conjecture that u-compactness of X for some u ∈ ω∗ (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction...