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E. G. Pytkeev 《Journal of Mathematical Sciences》2006,136(5):4131-4155
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).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 3–39, 2003. 相似文献
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Gary Gruenhage David Lutzer 《Proceedings of the American Mathematical Society》2000,128(10):3115-3124
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.
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J. Arias De Reyna 《Israel Journal of Mathematics》1982,42(1-2):33-36
We construct two prehilbertian Baire spaces whose product is not a Baire space.
An erratum to this article is available at . 相似文献
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Lá szló Zsilinszky 《Proceedings of the American Mathematical Society》1996,124(8):2575-2584
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.
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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. 相似文献
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Jiling Cao Salvador Garcí a-Ferreira Valentin Gutev 《Proceedings of the American Mathematical Society》2007,135(1):299-303
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.
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Stelios Negrepontis 《Archiv der Mathematik》1967,18(6):603-608
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. 相似文献
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E. G. Pytkeev 《Mathematical Notes》1992,52(3):953-959
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. 相似文献
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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
where osc stands for the supremum of the oscillations of at all points . As a consequence of the above we prove that for functions in two variables , X complete metric and K compact, there exists a G
δ-dense set such that the oscillation of f at each is bounded by the oscillations of the partial functions f
x
and f
k
. A representative result in this direction, that we prove using games, is the following: if X is a σ–β-unfavorable space and K is a compact space, then there exists a dense G
δ-subset D of X such that, for each ,
When the right hand side of the above inequality is zero we are dealing with separately continuous functions and we obtain as a particular case some well-known results obtained by the third named author in the mid 1970s.
C. Angosto, B. Cascales and I. Namioka are supported by the Spanish grants MTM2005-08379 (MEC & FEDER) and 00690/PI/04 (Fund.
Séneca). C. Angosto is also supported by the FPU grant AP2003-4443 (MEC & FEDER). 相似文献
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J. Ka?kol 《Journal of Mathematical Analysis and Applications》2007,332(2):965-974
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. 相似文献
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By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a Kσ 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. 相似文献
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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. 相似文献
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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. 相似文献
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Gerard J. Murphy 《Integral Equations and Operator Theory》1992,15(5):825-852
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. 相似文献
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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. 相似文献