297 research outputs found

    On disjoint Borel uniformizations

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    Larman showed that any closed subset of the plane with uncountable vertical cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that Larman's result is best possible: there exist closed sets with uncountable cross-sections which do not have more than aleph_1 disjoint Borel uniformizations, even if the continuum is much larger than aleph_1. This negatively answers some questions of Mauldin. The proof is based on a result of Stern, stating that certain Borel sets cannot be written as a small union of low-level Borel sets. The proof of the latter result uses Steel's method of forcing with tagged trees; a full presentation of this method, written in terms of Baire category rather than forcing, is given here

    Disjoint Infinity-Borel Functions

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    This is a followup to a paper by the author where the disjointness relation for definable functions from ωω{^\omega \omega} to ωω{^\omega \omega} is analyzed. In that paper, for each a∈ωωa \in {^\omega \omega} we defined a Baire class one function fa:ωω→ωωf_a : {^\omega \omega} \to {^\omega \omega} which encoded aa in a certain sense. Given g:ωω→ωωg : {^\omega \omega} \to {^\omega \omega}, let Ψ(g)\Psi(g) be the statement that gg is disjoint from at most countably many of the functions faf_a. We show the consistency strength of (∀g) Ψ(g)(\forall g)\, \Psi(g) is that of an inaccessible cardinal. We show that AD+\textrm{AD}^+ implies (∀g) Ψ(g)(\forall g)\, \Psi(g). Finally, we show that assuming large cardinals, (∀g) Ψ(g)(\forall g)\, \Psi(g) holds in models of the form L(R)[U]L(\mathbb{R})[\mathcal{U}] where U\mathcal{U} is a selective ultrafilter on ω\omega.Comment: 16 page

    Foliations with few non-compact leaves

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    Let F be a foliation of codimension 2 on a compact manifold with at least one non-compact leaf. We show that then F must contain uncountably many non-compact leaves. We prove the same statement for oriented p-dimensional foliations of arbitrary codimension if there exists a closed p form which evaluates positively on every compact leaf. For foliations of codimension 1 on compact manifolds it is known that the union of all non-compact leaves is an open set [A Haefliger, Varietes feuilletes, Ann. Scuola Norm. Sup. Pisa 16 (1962) 367-397].Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-12.abs.htm

    Essential countability of treeable equivalence relations

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    We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of E1 into E. Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one

    Analytic Colorings

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    We investigate the existence of perfect homogeneous sets for analytic colorings. An analytic coloring of X is an analytic subset of [X]^N, where N>1 is a natural number. We define an absolute rank function on trees representing analytic colorings, which gives an upper bound for possible cardinalities of homogeneous sets and which decides whether there exists a perfect homogeneous set. We construct universal sigma-compact colorings of any prescribed rank gamma<omega_1. These colorings consistently contain homogeneous sets of cardinality aleph_gamma but they do not contain perfect homogeneous sets. As an application, we discuss the so-called defectedness coloring of subsets of Polish linear spaces

    Locally Constant Functions

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    Let X be a compact Hausdorff space and M a metric space. E_0(X,M) is the set of f in C(X,M) such that there is a dense set of points x in X with f constant on some neighborhood of x. We describe some general classes of X for which E_0(X,M) is all of C(X,M). These include beta N - N, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case that M is a Banach space, we discuss the properties of E_0(X,M) as a normed linear space. We also build three first countable Eberlein compact spaces, F,G,H, with various E_0 properties: For all metric M: E_0(F,M) contains only the constant functions, and E_0(G,M) = C(G,M). If M is the Hilbert cube or any infinite dimensional Banach space, E_0(H,M) is not all of C(H,M), but E_0(H,M) = C(H,M) whenever M is a subset of RR^n for some finite n
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