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First countable spaces without point-countable π-bases

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (2007)

Fundamenta Mathematicae

We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that ∙ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable π-base (in fact, the minimum order of a π-base of the space can be made arbitrarily large); ∙ if there is a κ-Suslin line then there is a first countable GO-space of cardinality κ⁺ in which the order of any π-base is at least κ; ∙ it is consistent to have a first countable,...

Generalized Helly spaces, continuity of monotone functions, and metrizing maps

Lech Drewnowski, Artur Michalak (2008)

Fundamenta Mathematicae

Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes...

Generalized linearly ordered spaces and weak pseudocompactness

Oleg Okunev, Angel Tamariz-Mascarúa (1997)

Commentationes Mathematicae Universitatis Carolinae

A space X is truly weakly pseudocompact if X is either weakly pseudocompact or Lindelöf locally compact. We prove that if X is a generalized linearly ordered space, and either (i) each proper open interval in X is truly weakly pseudocompact, or (ii) X is paracompact and each point of X has a truly weakly pseudocompact neighborhood, then X is truly weakly pseudocompact. We also answer a question about weakly pseudocompact spaces posed by F. Eckertson in [Eck].

Hyperspace selections avoiding points

Valentin Gutev (2022)

Commentationes Mathematicae Universitatis Carolinae

We deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case --- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic...

In search for Lindelöf C p ’s

Raushan Z. Buzyakova (2004)

Commentationes Mathematicae Universitatis Carolinae

It is shown that if X is a first-countable countably compact subspace of ordinals then C p ( X ) is Lindelöf. This result is used to construct an example of a countably compact space X such that the extent of C p ( X ) is less than the Lindelöf number of C p ( X ) . This example answers negatively Reznichenko’s question whether Baturov’s theorem holds for countably compact spaces.

Infinite games and chain conditions

Santi Spadaro (2016)

Fundamenta Mathematicae

We apply the theory of infinite two-person games to two well-known problems in topology: Suslin’s Problem and Arhangel’skii’s problem on the weak Lindelöf number of the G δ topology on a compact space. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable, and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf...

Intersection topologies with respect to separable GO-spaces and the countable ordinals

M. Jones (1995)

Fundamenta Mathematicae

Given two topologies, T 1 and T 2 , on the same set X, the intersection topologywith respect to T 1 and T 2 is the topology with basis U 1 ∩ U 2 : U 1 ∈ T 1 , U 2 ∈ T 2 . Equivalently, T is the join of T 1 and T 2 in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and ω 1 -compactness in this class of topologies. We demonstrate that the majority of his results generalise...

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