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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.

We consider the question: when does a Ψ-space satisfy property (a)? We show that if $\left|A\right|<p$ then the Ψ-space Ψ(A) satisfies property (a), but in some Cohen models the negation of CH holds and every uncountable Ψ-space fails to satisfy property (a). We also show that in a model of Fleissner and Miller there exists a Ψ-space of cardinality $p$ which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).

We introduce a two player topological game and study the relationship of the existence of winning strategies to base properties and covering properties of the underlying space. The existence of a winning strategy for one of the players is conjectured to be equivalent to the space have countable network weight. In addition, connections to the class of D-spaces and the class of hereditarily Lindelöf spaces are shown.

Players ONE and TWO play the following game: In the nth inning ONE chooses a set $O_n$ from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset $T_n$ of X. The players must obey the rule that $O_n\subseteq O_{n+1}\subseteq T_{n+1}\subseteq T_n$ for each n. TWO wins if the intersection of TWO’s sets is equal to the union of ONE’s sets. If ONE has no winning strategy, then each element of ℱ is a $G_{\delta }$-set. To what extent is the converse true? We show that:  (A) For ℱ the collection of countable subsets of X:   1. There are subsets...

Natural weakenings of uniformizability of a ladder system on ω₁ are considered. It is shown that even assuming CH all the properties may be distinct in a strong sense. In addition, these properties are studied in conjunction with other properties inconsistent with full uniformizability, which we call anti-uniformization properties. The most important conjunction considered is the uniformization property we call countable metacompactness and the anti-uniformization property we call thinness. The...

Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology ${\tau }_{W\left(\rho \right)}$. It is known that $⟨CL\left(X\right),{\tau }_{W\left(\rho \right)}⟩$ is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $⟨CL\left(X\right),{\tau }_{W\left(\rho \right)}⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨CL\left(X\right),{\tau }_{W\left(\rho \right)}⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.

We construct from ⋄ a T₂ example of a hereditarily Lindelöf space X that is not a D-space but is the union of two subspaces both of which are D-spaces. This answers a question of Arhangel'skii.

We study the relation between the Hurewicz and Menger properties of filters considered topologically as subspaces of $𝒫\left(\omega \right)$ with the Cantor set topology.

The Katětov ordering of two maximal almost disjoint (MAD) families $𝒜$ and $ℬ$ is defined as follows: We say that $𝒜{\le }_Kℬ$ if there is a function $f:\omega \to \omega$ such that $f^{-1}\left(A\right)\in ℐ\left(ℬ\right)$ for every $A\in ℐ\left(𝒜\right)$. In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$-uniform if for every $X\in ℐ{\left(𝒜\right)}^{+}$, we have that ${𝒜|}_X{\le }_K𝒜$. We prove that CH implies that for every $K$-uniform MAD family $𝒜$ there is a $P$-point $p$ of ${\omega }^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense in the...

We show that if $𝒜$ is an uncountable AD (almost disjoint) family of subsets of $\omega$ then the space $\Psi \left(𝒜\right)$ does not admit a continuous selection; moreover, if $𝒜$ is maximal then $\Psi \left(𝒜\right)$ does not even admit a continuous selection on pairs, answering thus questions of T. Nogura.

We recall some classical results relating normality and some natural weakenings of normality in $\Psi$-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $\lambda$-sets and $\sigma$-sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being ${\aleph }_0$-separated. This new class fits between $\lambda$-sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $𝒜$ being potentially...