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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 dimension of X^n where X is a separable metric space

John Kulesza (1996)

Fundamenta Mathematicae

For a separable metric space X, we consider possibilities for the sequence S ( X ) = d n : n ∈ ℕ where d n = d i m X n . In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is X n such that S ( X n ) = n , n + 1 , n + 2 , . . . , Y n , for n >1, such that S ( Y n ) = n , n + 1 , n + 2 , n + 2 , n + 2 , . . . , and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of ℝ 2 is shown to exist which satisfies 1 = d i m X = d i m X 2 and d i m X 3 = 2 .

The fixed point set of open mappings on extremally disconnected spaces

Egbert Thümmel (1994)

Commentationes Mathematicae Universitatis Carolinae

We give an example of an extremally disconnected compact Hausdorff space with an open continuous selfmap such that the fixed point set is nonvoid and nowhere dense, respṫhat there is exactly one nonisolated fixed point.

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.

The union of two D-spaces need not be D

Dániel T. Soukup, Paul J. Szeptycki (2013)

Fundamenta Mathematicae

We construct from ⋄ a T₂ example of a hereditarily Lindelöf space X that is not a D-space but is the union of two subspaces both of which are D-spaces. This answers a question of Arhangel'skii.

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