A countably compact free Abelian group whose size has countable cofinality

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...

Forcing Minimal Extensions of Boolean Algebras

We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications: 1) We make Fedorchuk’s method more flexible, obtaining, for every cardinal λ of uncountable cofinality, a consistent example of a Boolean algebra Aλ whose every infinite homomorphic image is of cardin...

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