Countably compact, locally countable $T\sb{2}$-spaces

CH and first countable , countably compact spaces ✩

We show that it is consistent with the Continuum Hypothesis that first countable, countably compact spaces with no uncountable free sequences are compact. As a consequence, we get that CH does not imply the existence of a perfectly normal, countably compact, non-compact space, answering a question of Nyikos (Question 287 in the numbering of van Mill and Reed, Open Problems in Topology, Elsevier...

First Countable, Countably Compact Spaces and the Continuum Hypothesis

We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of ω1 with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah’s iteration theorems appearing in Chapters V and VIII of Proper an...

Locally Compact, Locally Countable Spaces and Random Reals

In this note I will present a proof that, assuming PFA, if R is a measure algebra then after forcing with R every uncountable locally compact locally countable cometrizable space contains an uncountable discrete set. The lemmas and techniques will be presented in a general form as they may be applicable to other problems.

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...

On first countable , countably compact spaces III : The problem of obtaining separable noncompact examples