Preimages of Baire spaces

Časopis pro pěstování matematiky

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Časopis pro pěstování matematiky, roč. 108 (1983), Praha

Proceedings of the American Mathematical Society, 2000

In this paper we describe broad classes of spaces for which the Baire space property is equivalent to the assertion that any two dense G δ -sets have dense intersection. We also provide examples of spaces where the equivalence does not hold. Finally, our techniques provide an easy proof of a new internal characterization of perfectly meager subspaces of [0, 1] and characterize metric spaces that are always of first category. MR classifications: Primary: 54E52; Secondary: 54E20, 54E25, 54E30, 54E35, 54H05, 54F65 Key words and phrases: Baire space, Volterra space, metric space, Moore space, Lasnev space, linearly ordered topological space, perfectly meager set, λ-set, always first category.

2005

In this paper, we study some properties of the Banach space β • α (X, E), consists of all Baire functions with relatively compact ranges from a perfectly normal space X into a Banach space E. Moreover, we establish that if β • α (X, E) is linear isometric with β • α (Y, E), then some compactification of X and Y are homeomorphic.

We prove that each coarsely homogenous separable metric space $X$ is coarsely equivalent to one of the spaces: the sigleton, the Cantor macro-cube or the Baire macro-space. This classification is derived from coarse characterizations of the Cantor macro-cube and of the Baire macro-space given in this paper. Namely, we prove that a separable metric space $X$ is coarsely equivalent to the Baire macro-space if any only if $X$ has asymptotic dimension zero and has unbounded geometry in the sense that for every $\delta$ there is $\epsilon$ such that no $\epsilon$-ball in $X$ can be covered by finitely many sets of diameter $\le \delta$.

Proceedings of the American Mathematical Society, 2006

We prove that if the Vietoris hyperspace CL(X) of all nonempty closed subsets of a space X is Baire, then all finite powers of X must be Baire spaces. In particular, there exists a metrizable Baire space X whose Vietoris hyperspace CL(X) is not Baire. This settles an open problem of R. A. McCoy stated in 1975.

Proceedings of the American Mathematical Society, 2008

Missouri Journal of Mathematical Sciences, 2010

In this paper, using bitopological semi-open sets, an asymmetric generalization of Haworth-McCoy's well-known theorem [4, Theorem 3.1] about Baire spaces is established.

Proceedings of the American Mathematical Society, 1990

We give functional characterizations of Baire spaces in the class of topological spaces with a σ \sigma -locally finite almost-base. This answers positively a question asked in [5] for topological spaces with a σ \sigma -locally finite almost-base.

Matematicki Vesnik

We prove that if X is a separable metric space with the Hurewicz covering property, then the Banach-Mazur game played on X is determined. The implication is not true when "Hurewicz covering property" is replaced with "Menger covering property".

Lecture Notes in Mathematics, 1991

Certain subclasses of B 1 (K), the Baire-1 functions on a compact metric space K, are defined and characterized. Some applications to Banach spaces are given. 0. Introduction. Let X be a separable infinite dimensional Banach space and let K denote its dual ball, Ba(X *), with the weak* topology. K is compact metric and X may be naturally identified with a closed subspace of C(K). X * * may also be identified with a closed subspace of A ∞ (K), the Banach space of bounded affine functions on K in the sup norm. Our general objective is to deduce information about the isomorphic structure of X or its subspaces from the topological nature of the functions F ∈ X * * ⊆ A ∞ (K). A classical example of this type of result is: X is reflexive if and only if X * * ⊂ C(K). A second example is the following theorem. (B 1 (K) is the class of bounded Baire-1 functions on K and DBSC(K) is the subclass of differences of bounded semicontinuous functions on K. The precise definitions appear below in §1.) We write Y ֒→ X if Y is isomorphic to a subspace of X. Theorem A. Let X be a separable Banach space and let K = Ba(X *) with the weak* topology. a) [35] ℓ 1 ֒→ X iff X * * \ B 1 (K) = ∅. b) [7] c 0 ֒→ X iff [X * * ∩ DBSC(K)] \ C(K) = ∅. Theorem A provides the motivation for this paper: What can be said about X if X * * ∩ [B 1 (K) \ DBSC(K)] = ∅? To study this problem we consider various subclasses of