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Countable Compact Scattered T₂ Spaces and Weak Forms of AC

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable. (3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space...

Countable compactness and p -limits

Salvador García-Ferreira, Artur Hideyuki Tomita (2001)

Commentationes Mathematicae Universitatis Carolinae

For ∅ ≠ M ⊆ ω * , we say that X is quasi M -compact, if for every f : ω → X there is p ∈ M such that f ¯ ( p ) ∈ X , where f ¯ is the Stone-Čech extension of f . In this context, a space X is countably compact iff X is quasi ω * -compact. If X is quasi M -compact and M is either finite or countable discrete in ω * , then all powers of X are countably compact. Assuming C H , we give an example of a countable subset M ⊆ ω * and a quasi M -compact space X whose square is not countably compact, and show that in a model of A. Blass and S. Shelah every quasi...

Countable dense homogeneity and λ-sets

Rodrigo Hernández-Gutiérrez, Michael Hrušák, Jan 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...

Countable dense homogeneous filters and the Menger covering property

Dušan Repovš, Lyubomyr Zdomskyy, Shuguo Zhang (2014)

Fundamenta Mathematicae

We present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hernández-Gutiérrez and Hrušák. The method of the proof also allows us to obtain for any n ∈ ω ∪ {∞} an n-dimensional metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.

Countable fan-tightness versus countable tightness

Aleksander V. Arhangel'skii, Angelo Bella (1996)

Commentationes Mathematicae Universitatis Carolinae

Countable tightness is compared to the stronger notion of countable fan-tightness. In particular, we prove that countable tightness is equivalent to countable fan-tightness in countably compact regular spaces, and that countable fan-tightness is preserved by pseudo-open compact mappings. We also discuss the behaviour of countable tightness and of countable fan-tightness under the product operation.

Countable products of Čech-scattered supercomplete spaces

Aarno Hohti, Zi Qiu Yun (1999)

Czechoslovak Mathematical Journal

We prove by using well-founded trees that a countable product of supercomplete spaces, scattered with respect to Čech-complete subsets, is supercomplete. This result extends results given in [Alstera], [Friedlera], [Frolika], [HohtiPelantb], [Pelanta] and its proof improves that given in [HohtiPelantb].

Countable products of spaces of finite sets

Antonio Avilés (2005)

Fundamenta Mathematicae

We consider the compact spaces σₙ(Γ) of subsets of Γ of cardinality at most n and their countable products. We give a complete classification of their Banach spaces of continuous functions and a partial topological classification.

Countable sums and products of Loeb and selective metric spaces

Horst Herrlich, Kyriakos Keremedis, Eleftherios Tachtsis (2005)

Commentationes Mathematicae Universitatis Carolinae

We investigate the role that weak forms of the axiom of choice play in countable Tychonoff products, as well as countable disjoint unions, of Loeb and selective metric spaces.

Countable tightness in the spaces of regular probability measures

Grzegorz Plebanek, Damian Sobota (2015)

Fundamenta Mathematicae

We prove that if K is a compact space and the space P(K × K) of regular probability measures on K × K has countable tightness in its weak* topology, then L₁(μ) is separable for every μ ∈ P(K). It has been known that such a result is a consequence of Martin's axiom MA(ω₁). Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todorčević on measures on Rosenthal compacta.

Countable Toronto spaces

Gary Gruenhage, J. Moore (2000)

Fundamenta Mathematicae

A space X is called an α-Toronto space if X is scattered of Cantor-Bendixson rank α and is homeomorphic to each of its subspaces of the same rank. We answer a question of Steprāns by constructing a countable α-Toronto space for each α ≤ ω. We also construct consistent examples of countable α-Toronto spaces for each α < ω 1 .

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