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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 box product of countably many metrizable spaces need not be normal

Eric van Douwen (1975)

Fundamenta Mathematicae

The insertion of $G_{δ}$ sets and fine topologies

Jaroslav Lukeš, Luděk Zajíček (1977)

Commentationes Mathematicae Universitatis Carolinae

The Lusin-Menchoff property of fine topologies

Jaroslav Lukeš (1977)

Commentationes Mathematicae Universitatis Carolinae

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.