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On the preservation of Baire and weakly Baire category

Alireza Kamel Mirmostafaee, Zbigniew Piotrowski (2016)

Mathematica Bohemica

We consider the question of preservation of Baire and weakly Baire category under images and preimages of certain kind of functions. It is known that Baire category is preserved under image of quasi-continuous feebly open surjections. In order to extend this result, we introduce a strictly larger class of quasi-continuous functions, i.e. the class of quasi-interior continuous functions. We show that Baire and weakly Baire categories are preserved under image of feebly open quasi-interior continuous...

On Typical Compact Convex Sets in Hilbert Spaces

De Blasi, F. (1997)

Serdica Mathematical Journal

Let E be an infinite dimensional separable space and for e ∈ E and X a nonempty compact convex subset of E, let qX(e) be the metric antiprojection of e on X. Let n ≥ 2 be an arbitrary integer. It is shown that for a typical (in the sence of the Baire category) compact convex set X ⊂ E the metric antiprojection qX(e) has cardinality at least n for every e in a dense subset of E.

On β-favorability of the strong Choquet game

László Zsilinszky (2011)

Colloquium Mathematicae

In the main result, partially answering a question of Telgársky, the following is proven: if X is a first countable R₀-space, then player β (i.e. the EMPTY player) has a winning strategy in the strong Choquet game on X if and only if X contains a nonempty W δ -subspace which is of the first category in itself.

On ω -resolvable and almost- ω -resolvable spaces

J. Angoa, M. Ibarra, Angel Tamariz-Mascarúa (2008)

Commentationes Mathematicae Universitatis Carolinae

We continue the study of almost- ω -resolvable spaces beginning in A. Tamariz-Mascar’ua, H. Villegas-Rodr’ıguez, Spaces of continuous functions, box products and almost- ω -resoluble spaces, Comment. Math. Univ. Carolin. 43 (2002), no. 4, 687–705. We prove in ZFC: (1) every crowded T 0 space with countable tightness and every T 1 space with π -weight ≤ ℵ 1 is hereditarily almost- ω -resolvable, (2) every crowded paracompact T 2 space which is the closed preimage of a crowded Fréchet T 2 space in such a way that the...

Ordered Cauchy spaces.

Kent, Darrell C., Vainio, R. (1985)

International Journal of Mathematics and Mathematical Sciences

Points with maximal Birkhoff average oscillation

Jinjun Li, Min Wu (2016)

Czechoslovak Mathematical Journal

Let f : X → X be a continuous map with the specification property on a compact metric space X . We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller and locally...

Préimages d’espaces héréditairement de Baire

Ahmed Bouziad (1997)

Fundamenta Mathematicae

The main result is slightly more general than the following statement: Let f: X → Y be a quasi-perfect mapping, where X is a regular space and Y a Hausdorff totally non-meagre space; if X or Y is χ-scattered, or if Y is a Lasnev space, then X is totally non-meagre. In particular, the product of a compact space X and a Hausdorff regular totally non-meagre space Y which is χ-scattered or a Lasnev space, is totally non-meagre.

Preimages of Baire spaces

Jozef Doboš, Zbigniew Piotrowski, Ivan L. Reilly (1994)

Mathematica Bohemica

A simple machinery is developed for the preservation of Baire spaces under preimages. Subsequently, some properties of maps which preserve nowhere dense sets are given.

Products of Baire spaces revisited

László Zsilinszky (2004)

Fundamenta Mathematicae

Generalizing a theorem of Oxtoby, it is shown that an arbitrary product of Baire spaces which are almost locally universally Kuratowski-Ulam (in particular, have countable-in-itself π-bases) is a Baire space. Also, partially answering a question of Fleissner, it is proved that a countable box product of almost locally universally Kuratowski-Ulam Baire spaces is a Baire space.

Products, the Baire category theorem, and the axiom of dependent choice

Horst Herrlich, Kyriakos Keremedis (1999)

Commentationes Mathematicae Universitatis Carolinae

In ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the following statements are shown to be equivalent: (i) The axiom of dependent choice. (ii) Products of compact Hausdorff spaces are Baire. (iii) Products of pseudocompact spaces are Baire. (iv) Products of countably compact, regular spaces are Baire. (v) Products of regular-closed spaces are Baire. (vi) Products of Čech-complete spaces are Baire. (vii) Products of pseudo-complete spaces are Baire.

Relaciones entre las compleciones de espacios seudométricos y espacios uniformes.

Juan Margalef Roig, Enrique Outerelo Domínguez (1979)

Revista Matemática Hispanoamericana

En el párrafo 1 se describe el isomorfismo uniforme existente entre la compleción de Hausdorff de un espacio métrico y la compleción de Weil del espacio uniforme separado asociado al espacio métrico.En el párrafo 2 se construye la compleción reducida de un espacio seudométrico y se describe el isomorfismo uniforme existente entre la compleción reducida de un espacio seudométrico y la compleción reducida del espacio uniforme asociado al espacio seudométrico.Finalmente, a lo largo del párrafo 3, se...

Rosenthal compacta and NIP formulas

Pierre Simon (2015)

Fundamenta Mathematicae

We apply the work of Bourgain, Fremlin and Talagrand on compact subsets of the first Baire class to show new results about ϕ-types for ϕ NIP. In particular, we show that if M is a countable model, then an M-invariant ϕ-type is Borel-definable. Also, the space of M-invariant ϕ-types is a Rosenthal compactum, which implies a number of topological tameness properties.

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