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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...