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Spaces with large relative extent

Yan-Kui Song (2007)

Czechoslovak Mathematical Journal

In this paper, we prove the following statements: (1) For every regular uncountable cardinal κ , there exist a Tychonoff space X and Y a subspace of X such that Y is both relatively absolute star-Lindelöf and relative property (a) in X and e ( Y , X ) ≥ κ , but Y is not strongly relative star-Lindelöf in X and X is not star-Lindelöf. (2) There exist a Tychonoff space X and a subspace Y of X such that Y is strongly relative star-Lindelöf in X (hence, relative star-Lindelöf), but Y is not absolutely relative star-Lindelöf...

Special sets of reals and weak forms of normality on Isbell--Mrówka spaces

Vinicius de Oliveira Rodrigues, Victor dos Santos Ronchim, Paul J. Szeptycki (2023)

Commentationes Mathematicae Universitatis Carolinae

We recall some classical results relating normality and some natural weakenings of normality in Ψ -spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like Q -sets, λ -sets and σ -sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being ℵ 0 -separated. This new class fits between λ -sets and perfectly meager sets. We also discuss conditions for an almost disjoint family 𝒜 being potentially...

The Arkhangel’skiĭ–Tall problem: a consistent counterexample

Gary Gruenhage, Piotr Koszmider (1996)

Fundamenta Mathematicae

We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in [ ω ] ω , and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.

The Arkhangel'skiĭ–Tall problem under Martin’s Axiom

Gary Gruenhage, Piotr Koszmider (1996)

Fundamenta Mathematicae

We show that MA σ - c e n t e r e d ( ω 1 ) implies that normal locally compact metacompact spaces are paracompact, and that MA( ω 1 ) implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.

The Niemytzki plane is ϰ -metrizable

Wojciech Bielas, Andrzej Kucharski, Szymon Plewik (2021)

Mathematica Bohemica

We prove that the Niemytzki plane is ϰ -metrizable and we try to explain the differences between the concepts of a stratifiable space and a ϰ -metrizable space. Also, we give a characterisation of ϰ -metrizable spaces which is modelled on the version described by Chigogidze.

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