Baire functions and spaces of Baire functions

The paper considers (1) the tightness of spaces of Baire functions and their subspaces endowed with the topology of pointwise convergence; (2) Z _σ-mappings of K-analytic spaces; (3) K _σ-analytic spaces (Tychonoff spaces that are Z _σ-images of K-analytic spaces). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 3–39, 2003.     

Baire and Volterra spaces

In this paper we describe broad classes of spaces for which the Baire space property is equivalent to the assertion that any two dense -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 and characterize metric spaces that are always of first category.

     

Normed barely Baire spaces

We construct two prehilbertian Baire spaces whose product is not a Baire space. An erratum to this article is available at .     

Baire spaces and hyperspace topologies

Sufficient conditions for abstract (proximal) hit-and-miss hyperspace topologies and the Wijsman hyperspace topology, respectively, are given to be Baire spaces, thus extending results of McCoy, Beer, and Costantini. Further the quasi-regularity of (proximal) hit-and-miss topologies is investigated.

     

Baire spaces and infinite games

It is well known that if the nonempty player of the Banach–Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box product topology. The converse of this implication may also be true: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.     

Baire spaces and Vietoris hyperspaces

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

     

Baire sets in topological spaces

Summary In this paper we study topological properties of Baire sets in various classes of spaces. The main results state that a Baire set in a realcompact space is realcompact; a Baire set in a topologically complete space is topologically complete; and that a pseudocompact Baire set in any topological space is a zero-set. As a consequence, we obtain new characterizations of realcompact and pseudocompact spaces in terms of Baire sets of their Stone-ech compactifications. (Lorch in [3] using a different method has obtained either implicitly or explicitly the same results concerning Baire sets in realcompact spaces.) The basic tools used for these proofs are first, the notions of anr-compactification andr-embedding (see below for definitions), which have also been defined and used independently byMrówka in [4]; second, the idea included in the proof of the theorem: Every compact Baire set is aG as given inHalmos' text on measure theory [2; Section 51, theorem D].The author wishes to thank Professor W. W.Comfort for his valuable advice in the preparation of this paper.     

On spaces of Baire I functions over K-analytic spaces

Suppose that is a relatively countably compact subset of B₁(X), the space of Baire I functions over a K-analytic space X equipped with the pointwise convergence topology. It is proved that (1) the closure of is a strongly countably compact Frechét-Urysohn space; (2) if is ₁ -compact, is a bicompactum; (3) if X is a paracompact space, the closure of is a bicompactum.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 108–116, September, 1992.     

Distances to spaces of Baire one functions

Given a metric space X and a Banach space (E, ||·||) we use an index of σ-fragmentability for maps to estimate the distance of f to the space B ₁(X, E) of Baire one functions from X into (E, ||·||). When X is Polish we use our estimations for these distances to give a quantitative version of the well known Rosenthal’s result stating that in the pointwise relatively countably compact sets are pointwise relatively compact. We also obtain a quantitative version of a Srivatsa’s result that states that whenever X is metric any weakly continuous function belongs to B ₁(X, E): our result here says that for an arbitrary we have

Compact coverings for Baire locally convex spaces

Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space Cp(X) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff Cp(X) is Baire (i.e. of the second Baire category) and is covered by a family of compact sets such that Kα⊂Kβ if α?β. Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete.     

The Hurewicz dichotomy for generalized Baire spaces

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a K_σ subset of X if and only if it does not contain a closed-in-X subset homeomorphic to the Baire space ^ww. We consider the analogous statement (which we call the Hurewicz dichotomy) for ∑₁¹j subsets of the generalized Baire space ^κκ for a given uncountable cardinal κ with κ = κ^<κ. We show that the statement that this dichotomy holds at all uncountable regular cardinals is consistent with the axioms of ZFC together with GCH and large cardinal axioms. In contrast, we show that the dichotomy fails at all uncountable regular cardinals after we add a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the κ-perfect set property, the κ-Miller measurability, and the κ-Sacks measurability.     

Mappings of Baire spaces into function spaces and Kadeč renorming

Assuming that there exists in the unit interval [0, 1] a coanalytic set of continuum cardinality without any perfect subset, we show the existence of a scattered compact Hausdorff spaceK with the following properties: (i) For each continuous mapf on a Baire spaceB into (C(K), pointwise), the set of points of continuity of the mapf: B → (C(K), norm) is a denseG _δ subset ofB, and (ii)C(K) does not admit a Kadeč norm that is equivalent to the supremum norm. This answers the question of Deville, Godefroy and Haydon under the set theoretic assumption stated above.     

Baire generalized topological spaces, generalized metric spaces and infinite games

Different generalizations of topological Baire spaces to the case of generalized topological spaces are considered and the properties of such spaces are examined. In particular, these considerations are focused on the relationship between Baire generalized topological spaces and semi-continuous real functions and infinite games. The notion of generalized metric spaces corresponding to generalized topological spaces is introduced as an important tool in this discussion.     

Toeplitz operators on generalised H2 spaces

Some aspects are developed of the theory of Toeplitz operators on generalised HardyH ² spaces associated to function algebras. It is shown that a substantial number of results of the classical theory of Toeplitz operators on the circle extend to this situation, although counterexamples are given which show that there are also important differences. Spectral connectedness results are obtained, and a characterisation of invertibility for Toeplitz operators. The Fredholm theory is also studied.     

Baire number of the spaces of uniform ultrafilters

The Baire number is defined for a topological space without isolated points as the minimal size of the family of nowhere dense sets covering the space in question. We prove that in the case ofU(κ), the space of uniform ultrafilters over uncountable κ, the Baire number equals eitherω ₁ orω ₂, depending on the cofinality of κ. The results are connected to the collapsing of cardinals when using the quotient algebraP(κ) mod[κ]^<κ as the notion of forcing. The main portion of the present research, was done at the Center for Theoretical Study at Charles University and the Academy of Sciences.