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 $\u27e8CL\left(X\right),{\tau}_{W\left(\rho \right)}\u27e9$ 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 $\u27e8CL\left(X\right),{\tau}_{W\left(\rho \right)}\u27e9$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $\u27e8CL\left(X\right),{\tau}_{W\left(\rho \right)}\u27e9$ 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.

The Katětov ordering of two maximal almost disjoint (MAD) families $\mathcal{A}$ and $\mathcal{B}$ is defined as follows: We say that $\mathcal{A}{\le}_{K}\mathcal{B}$ if there is a function $f:\omega \to \omega $ such that ${f}^{-1}\left(A\right)\in \mathcal{I}\left(\mathcal{B}\right)$ for every $A\in \mathcal{I}\left(\mathcal{A}\right)$. 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 $X\in \mathcal{I}{\left(\mathcal{A}\right)}^{+}$, we have that ${\mathcal{A}|}_{X}{\le}_{K}\mathcal{A}$. We prove that CH implies that for every $K$-uniform MAD family $\mathcal{A}$ there is a $P$-point $p$ of ${\omega}^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense in the...

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

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

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