Metric spaces in which Blumberg’s theorem holds

Bradford, James C.; Goffman, Casper

doi:10.1090/s0002-9939-1960-0146310-1

Cited by 20 publications

(9 citation statements)

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“…This is analogous to Theorem II of [2], the Theorem of [1], and Lemma 3 of [4], with "FC" replaced by "T0". Since we have Lemma 2 above, we can follow the proof of Lemma 3 of [4]. Let {Gn} be a basis for R. For every n, let Yn = f~l(Gn), let Qn be the union of all open subsets of X in which Yn is non-/T0 dense, and let En = Y" -Q".…”

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confidence: 89%

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Variations on Lusin’s theorem

Brown

Prikry²

1987

Trans. Amer. Math. Soc.

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Abstract.We prove a theorem about continuous restrictions of Marczewski measurable functions to large sets. This theorem is closely related to the theorem of Lusin about continuous restrictions of Lebesgue measurable functions to sets of positive measure and the theorem of Nikodym and Kuratowski about continuous restrictions of functions with the Baire property (in the wide sense) to residual sets. This theorem is used to establish Lusin-type theorems for universally measurable functions and functions which have the Baire property in the restricted sense. The theorems are shown (under assumption of the Continuum Hypothesis) to be "best possible" within a certain context.1. Measurable functions. The results which appear here were announced at the Ninth Summer Symposium on Real Analysis, held at the University of Louisville during 1985, and appeared in abstract form (without proofs) in [9].We study theorems about functions from a complete metric space X without isolated points (or else from the unit interval I = [0,1]) into the reals R. d will denote the metric for the space X. c denotes the cardinality of the continuum, and CH refers to the Continuum Hypotheses. A perfect set is a closed set M such that every point of M is a limit point of M, and a Cantor set is a homeomorphic image of the " middle thirds" Cantor subset of I.The measurable functions we will be interested in are defined in terms of the following a-algebras of subsets of X:Bw: Baire property in the wide sense [22], Br: Baire property in the restricted sense [22], L: Lebesgue measurable sets (assuming X = I), U: Universally measurable sets (a set M is universally measurable if it is measurable with respect to the completion of every Borel measure on X), (s): Marczewski measurable sets (a set M is Marczewski measurable provided that for every perfect subset P of X, there exists a perfect subset Q of P which either misses M or is a subset of M), and B: Borel measurable.

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confidence: 89%

“…These are analogous to the lemmas concerning FC sets and arbitrary functions which are usually employed in proving variations on Blumberg's theorem about continuous restrictions of arbitrary functions (see [1][2][3][4][5][6][7][8] and [33][34]). Proof.…”

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confidence: 96%

Variations on Lusin’s theorem

Brown

Prikry²

1987

Trans. Amer. Math. Soc.

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“…It is clear from the proof of 1.1 given in [3], that any topological space for which 1.2 holds is a Baire space. The purposes of this note are to show that 1.2 holds for certain classes of topological Baire spaces, and to give an example which shows that 1.2 does not hold for all completely regular, Hausdorff, Baire spaces.…”

Section: Theoremmentioning

confidence: 99%

“…Blumberg proved that, if/is a real-valued function defined on the real line R, then there is a dense subset D of R such that/|D is continuous. J. C. Bradford and C. Goffman showed [3] that this theorem holds for a metric space X if and only if X is a Baire space. In the present paper, we show that Blumberg's theorem holds for a topological space X having a rr-disjoint pseudo-base if and only if X is a Baire space.…”

Section: Introductionmentioning

confidence: 99%

Topological spaces in which Blumberg’s theorem holds

White¹

1974

Proc. Amer. Math. Soc.

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Abstract.H. Blumberg proved that, if/is a real-valued function defined on the real line R, then there is a dense subset D of R such that/|D is continuous. J. C. Bradford and C. Goffman showed [3] that this theorem holds for a metric space X if and only if X is a Baire space. In the present paper, we show that Blumberg's theorem holds for a topological space X having a rr-disjoint pseudo-base if and only if X is a Baire space. Then we identify some classes of topological spaces which have (/-disjoint pseudo-bases. Also, we show that a certain class of iocally compact, Hausdorff spaces satisfies Blumberg's theorem. Finally, we describe two Baire spaces for which Blumberg's theorem does not hold. One is completely regular, Hausdorff, cocompact, strongly a-favorable, and pseudocomplete; the other is regular and hereditarily Lindelöf.1. In [3], J. C. Bradford and C. Goffman proved the following statement. Theorem.A metric space X is a Baire space if and only if the following statement, called Blumberg's theorem, holds.1.2. Iff is a real-valued function defined on X, then there is a dense subset D of Xsuch that f\D is continuous.It is clear from the proof of 1.1 given in [3], that any topological space for which 1.2 holds is a Baire space. The purposes of this note are to show that 1.2 holds for certain classes of topological Baire spaces, and to give an example which shows that 1.2 does not hold for all completely regular, Hausdorff, Baire spaces.1.3. Lemma. Suppose X and Y are topological spaces and f.X-^-Y. Suppose that for each nonempty open subset U of X, there is a subset KÍU) of U such that f\KÍU) is continuous and K{U) is not nowhere dense. Then there is a dense subset D of X such that f\D is continuous.The proof of 1.3 is simple and is omitted.

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“…In [4] J. C. Bradford and C. Goffman proved that every Blumberg space is Baire and in [14] R. Levy showed that there is consistently a compact Hausdorff, and therefore Baire, space which is not Blumberg.…”

Section: Spaces Of Countable Pseudocharacter and Blumberg Spacesmentioning

confidence: 99%