We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable. (3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space...

For $\varnothing \ne M\subseteq {\omega }^{*}$, we say that $X$ is quasi $M$-compact, if for every $f:\omega \to X$ there is $p\in M$ such that $\overline{f}\left(p\right)\in X$, where $\overline{f}$ is the Stone-Čech extension of $f$. In this context, a space $X$ is countably compact iff $X$ is quasi ${\omega }^{*}$-compact. If $X$ is quasi $M$-compact and $M$ is either finite or countable discrete in ${\omega }^{*}$, then all powers of $X$ are countably compact. Assuming $CH$, we give an example of a countable subset $M\subseteq {\omega }^{*}$ and a quasi $M$-compact space $X$ whose square is not countably compact, and show that in a model of A. Blass and S. Shelah every quasi...

We consider the compact spaces σₙ(Γ) of subsets of Γ of cardinality at most n and their countable products. We give a complete classification of their Banach spaces of continuous functions and a partial topological classification.

In this work we study countably z-compact spaces and z-Lindelof spaces. Several new properties of them are given. It is proved that every countably z-compact space is pseuodocompact (a space on which every real valued continuous function is bounded). Spaces which are countably z-compact but not countably compact are given. It is proved that a space is countably z-compact iff every countable z-closed set is compact. Characterizations of countably z-compact and z-Lindelof spaces by multifunctions...

For a transfinite cardinal κ and i ∈ 0,1,2 let ${ℒ}_i\left(\kappa \right)$ be the class of all linearly ordered spaces X of size κ such that X is totally disconnected when i = 0, the topology of X is generated by a dense linear ordering of X when i = 1, and X is compact when i = 2. Thus every space in ℒ₁(κ) ∩ ℒ₂(κ) is connected and hence ℒ₁(κ) ∩ ℒ₂(κ) = ∅ if $\kappa <2^{\aleph ₀}$, and ℒ₀(κ) ∩ ℒ₁(κ) ∩ ℒ₂(κ) = ∅ for arbitrary κ. All spaces in ℒ₁(ℵ₀) are homeomorphic, while ℒ₂(ℵ₀) contains precisely ℵ₁ spaces up to homeomorphism. The class ℒ₁(κ)...