Fragmentability and compactness in C(K)-spaces

B. Cascales; G. Manjabacas; G. Vera

Displaying similar documents to “Fragmentability and compactness in C(K)-spaces”

Functional separability

Ronnie Levy, M. Matveev (2010)

Commentationes Mathematicae Universitatis Carolinae

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A space $X$ is functionally countable (FC) if for every continuous $f : X \to ℝ$ , $| f (X) | \leq ω$ . The class of FC spaces includes ordinals, some trees, compact scattered spaces, Lindelöf P-spaces, $σ$ -products in $2^{κ}$ , and some L-spaces. We consider the following three versions of functional separability: $X$ is 1-FS if it has a dense FC subspace; $X$ is 2-FS if there is a dense subspace $Y \subset X$ such that for every continuous $f : X \to ℝ$ , $| f (Y) | \leq ω$ ; $X$ is 3-FS if for every continuous $f : X \to ℝ$ , there is a dense subspace $Y \subset X$ such that $| f (Y) | \leq ω$ . We give examples...

The Baire property in remainders of topological groups and other results

Aleksander V. Arhangel'skii (2009)

Commentationes Mathematicae Universitatis Carolinae

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It is established that a remainder of a non-locally compact topological group $G$ has the Baire property if and only if the space $G$ is not Čech-complete. We also show that if $G$ is a non-locally compact topological group of countable tightness, then either $G$ is submetrizable, or $G$ is the Čech-Stone remainder of an arbitrary remainder $Y$ of $G$ . It follows that if $G$ and $H$ are non-submetrizable topological groups of countable tightness such that some remainders of $G$ and $H$ are homeomorphic, then...

On a classification of pointwise compact sets of the first Baire class functions

Witold Marciszewski (1989)

Fundamenta Mathematicae

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Convex functions with non-Borel set of Gâteaux differentiability points

Petr Holický, M. Šmídek, Luděk Zajíček (1998)

Commentationes Mathematicae Universitatis Carolinae

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We show that on every nonseparable Banach space which has a fundamental system (e.gȯn every nonseparable weakly compactly generated space, in particular on every nonseparable Hilbert space) there is a convex continuous function $f$ such that the set of its Gâteaux differentiability points is not Borel. Thereby we answer a question of J. Rainwater (1990) and extend, in the same time, a former result of M. Talagrand (1979), who gave an example of such a function $f$ on $ℓ^{1} (𝔠)$ .

Space of Baire functions. I

J. E. Jayne (1974)

Annales de l'institut Fourier

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Several equivalent conditions are given for the existence of real-valued Baire functions of all classes on a type of $K$ -analytic spaces, called disjoint analytic spaces, and on all pseudocompact spaces. The sequential stability index for the Banach space of bounded continuous real-valued functions on these spaces is shown to be either $0, 1$ , or $Ω$ (the first uncountable ordinal). In contrast, the space of bounded real-valued Baire functions of class 1 is shown to contain closed linear subspaces...