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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.

MAD families and P -points

Salvador García-FerreiraPaul J. Szeptycki — 2007

Commentationes Mathematicae Universitatis Carolinae

The Katětov ordering of two maximal almost disjoint (MAD) families 𝒜 and is defined as follows: We say that 𝒜 K if there is a function f : ω ω such that f - 1 ( A ) ( ) for every A ( 𝒜 ) . 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 ( 𝒜 ) + , we have that 𝒜 | X K 𝒜 . We prove that CH implies that for every K -uniform MAD family 𝒜 there is a P -point p of ω * such that the set of all Rudin-Keisler predecessors of p is dense in the...

Selections on Ψ -spaces

Michael HrušákPaul J. SzeptyckiArtur Hideyuki Tomita — 2001

Commentationes Mathematicae Universitatis Carolinae

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

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