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...

Countable tightness is compared to the stronger notion of countable fan-tightness. In particular, we prove that countable tightness is equivalent to countable fan-tightness in countably compact regular spaces, and that countable fan-tightness is preserved by pseudo-open compact mappings. We also discuss the behaviour of countable tightness and of countable fan-tightness under the product operation.

We investigate the role that weak forms of the axiom of choice play in countable Tychonoff products, as well as countable disjoint unions, of Loeb and selective metric spaces.

A space X is called an α-Toronto space if X is scattered of Cantor-Bendixson rank α and is homeomorphic to each of its subspaces of the same rank. We answer a question of Steprāns by constructing a countable α-Toronto space for each α ≤ ω. We also construct consistent examples of countable α-Toronto spaces for each $\alpha <{\omega }_1$.

We present a construction from ♢* of a first countable, regular, countably metacompact space with a closed discrete subspace that is not a $G_{\delta }$. In addition some nonperfect spaces with σ-disjoint bases are constructed.

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ${}^{\omega }\omega$ of irrationals, or certain of its subspaces. In particular, given $f\in {}^{\omega }(\omega \setminus \left\{0\right\})$, we consider compact sets of the form ${\prod }_{i\in \omega }{B}_i$, where $|B_i|=f\left(i\right)$ for all, or for infinitely many, $i$. We also consider “$n$-splitting” compact sets, i.e., compact sets $K$ such that for any $f\in K$ and $i\in \omega$, $\left|\right\{g\left(i\right):g\in K,g\upharpoonright i=f\upharpoonright i\left\}\right|=n$.

We consider the space $D(X,Y)$ of densely continuous forms introduced by Hammer and McCoy and investigated also by Holá . We show some additional properties of $D(X,Y)$ and investigate the subspace $D^{*}\left(X\right)$ of locally bounded real-valued densely continuous forms equipped with the topology of pointwise convergence ${\tau }_p$. The largest part of the paper is devoted to the study of various cardinal functions for $(D^{*}\left(X\right),{\tau }_p)$, in particular: character, pseudocharacter, weight, density, cellularity, diagonal degree, $\pi$-weight, $\pi$-character,...