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Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-GutiérrezPaul J. Szeptycki — 2015

Fundamenta Mathematicae

Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology τ W ( ρ ) . It is known that C L ( X ) , τ W ( ρ ) 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 C L ( X ) , τ W ( ρ ) being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then C L ( X ) , τ W ( ρ ) is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.

Disconnectedness properties of hyperspaces

Rodrigo Hernández-GutiérrezAngel Tamariz-Mascarúa — 2011

Commentationes Mathematicae Universitatis Carolinae

Let X be a Hausdorff space and let be one of the hyperspaces C L ( X ) , 𝒦 ( X ) , ( X ) or n ( X ) ( n a positive integer) with the Vietoris topology. We study the following disconnectedness properties for : extremal disconnectedness, being a F ' -space, P -space or weak P -space and hereditary disconnectedness. Our main result states: if X is Hausdorff and F 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 ( X ) is hereditarily disconnected. We also...

Countable dense homogeneity and λ-sets

Rodrigo Hernández-GutiérrezMichael HrušákJan van Mill — 2014

Fundamenta Mathematicae

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...

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