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We prove that if ℱ is a non-meager P-filter, then both ℱ and ${}^{\omega }ℱ$ are countable dense homogeneous spaces.

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.

Let $X$ be a Hausdorff space and let $\mathscr{H}$ be one of the hyperspaces $CL\left(X\right)$, $𝒦\left(X\right)$, $ℱ\left(X\right)$ or ${ℱ}_n\left(X\right)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathscr{H}$: extremal disconnectedness, being a $F^{\text{'}}$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that (a) both $F$ and $X-F$ are totally disconnected, (b) the quotient $X/F$ is hereditarily disconnected, then $𝒦\left(X\right)$ is hereditarily disconnected. We also...

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.

We show that all sufficiently nice λ-sets are countable dense homogeneous (𝖢𝖣𝖧). From this fact we conclude that for every uncountable cardinal κ ≤ 𝔟 there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every 𝖢𝖣𝖧 metric space has size either ω₁ or 𝔠. An...