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In [M.R. Burke, Large entire cross-sections of second category sets in Rn+1Rn+1, Topology Appl. 154 (2007) 215–240], a model was constructed in which for any everywhere second category set A⊆Rn+1A⊆Rn+1 there is an entire function f:Rn→Rf:Rn→R which cuts a large section through A in the sen...
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By the Kuratowski-Ulam theorem, if A subset of Rn+1 = R-n x R is a Borel set which has second category intersection with every ball (i.e., is "everywhere second category"), then there is a y is an element of R such that the section A boolean AND (R-n x {y}) is everywhere second category in R-n x {y}...
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In this paper we investigate two main problems. One of them is the question on the existence of category liftings in the product of two topological spaces. We prove, that if X x Y is a Baire space, then, given (strong) category liftings rho and sigma on X and Y, respectively, there exists a (strong)...
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We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A boolean AND C is nonmeager in C. We also examine variants of this result and establish a measure theoretic analog.
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We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-iso...
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This paper continues the investigation begun in [M.R. Burke, Topology Appl. 129 (2003) 29-65] into the measurability properties of separately continuous functions. We sharpen several results from that paper. (1) If X is any product of countably compact Dedekind complete linearly ordered spaces, then...
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We study the class of Tychonoff spaces that can be mapped continuously into R in such a way that the preimage of every nowhere dense set is nowhere dense. We show that every metric space without isolated points is in this class. We also give examples of spaces which have nowhere constant continuous ...
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Diamond, Pomerance and Rubel (1981) proved that there are subsets M of the complex plane such that for any two entire functions f and g if f[M] = g[M], then f = g. Baraducci and Dikranjan showed in 1993 that the continuum hypothesis (CH) implies the existence of a similar set M subset of R for the c...
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We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range.
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The purpose of this paper is to determine which metric spaces X and Y are such that the uniformly continuous maps f : X --> Y are precisely the continuous maps between (X, tau1) and (Y, tau2) for some new topologies tau1 and tau2 on X and Y respectively.
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The current state of knowledge concerning liftings for noncomplete probability spaces is discussed. This is a somewhat expanded version of the author's talk given at the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work.
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For each locally compact group G with Haar measure mu , we obtain the following results. The first is a version for group quotients of a classical result of Kuratowski and Ulam on first category subsets of the plane. The second is a strengthening of a theorem of Kupka and Prikry; we obtain it by a m...
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We show that it is consistent with ZFC that L(infinity)(Y, B, upsilon) has no linear lifting for many non-complete probability spaces (Y, B, upsilon), in particular for Y = [0, 1]A, B = Borel subsets of Y, upsilon = usual Radon measure on B.
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We call E subset-of-or-is-equal-to {0,1}kappa projective if for some countable A subset-of-or-is-equal-to kappa there is an E(A) subset-of-or-is-equal-to {0,1}A such that E = E(A) x {0,1}kappa/A and E(A) is a projective subset of the Cantor set {0,1}A. We construct a model where Haar measure on {0,1...
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We investigate the cofinality of the partial order N-kappa of functions from a regular cardinal kappa into the ideal N of Lebesgue measure zero subsets of R. We show that when add(N) = kappa and the covering lemma holds with respect to an inner model of GCH, then cf(N-kappa) = max{cf(kappa(kappa)), ...
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This is a survey paper giving a self-contained account of Shelah's theory of the pcf function pcf(a) = {cf(PI a/D, < D): D is an ultrafilter on a}, where a is a set of regular cardinals such that \a\ < min(a). We also give several applications of the theory to cardinal arithmetic, the exi...
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