77 research outputs found

    Covering an uncountable square by countably many continuous functions

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    We prove that there exists a countable family of continuous real functions whose graphs together with their inverses cover an uncountable square, i.e. a set of the form X×XX\times X, where XX is an uncountable subset of the real line. This extends Sierpi\'nski's theorem from 1919, saying that S×SS\times S can be covered by countably many graphs of functions and inverses of functions if and only if the size of SS does not exceed 1\aleph_1. Our result is also motivated by Shelah's study of planar Borel sets without perfect rectangles.Comment: Added new results (9 pages

    A five element basis for the uncountable linear orders

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    In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are X, omega_1, omega_1^*, C, C^* where X is any suborder of the reals of cardinality aleph_1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah.Comment: 21 page

    The embedding structure for linearly ordered topological spaces

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    In this paper, the class of all linearly ordered topological spaces (LOTS) quasi-ordered by the embeddability relation is investigated. In ZFC it is proved that for countable LOTS this quasi-order has both a maximal (universal) element and a finite basis. For the class of uncountable LOTS of cardinality κ\kappa it is proved that this quasi-order has no maximal element for κ\kappa at least the size of the continuum and that in fact the dominating number for such quasi-orders is maximal, i.e. 2κ2^\kappa. Certain subclasses of LOTS, such as the separable LOTS, are studied with respect to the top and internal structure of their respective embedding quasi-order. The basis problem for uncountable LOTS is also considered; assuming the Proper Forcing Axiom there is an eleven element basis for the class of uncountable LOTS and a six element basis for the class of dense uncountable LOTS in which all points have countable cofinality and coinitiality

    On what I do not understand (and have something to say): Part I

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    This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (``see ... '' means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The other half, concentrating on model theory, will subsequently appear

    Algebraic properties of profinite groups

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    Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update

    Characterization and Transformation of Countryman Lines and R-Embeddable Coherent Trees in ZFC

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    Ph.DDOCTOR OF PHILOSOPH

    Category forcings, MM+++MM^{+++}, and generic absoluteness for the theory of strong forcing axioms

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    We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces MM++MM^{++} and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of MM++MM^{++} collapsing a superhuge cardinal to become the second uncountable cardinal produces a model in which the cutoff of the category of stationary set preserving forcings at any rank initial segment of the universe of large enough height is forcing equivalent to a presaturated tower of normal filters. We let MM+++MM^{+++} denote this statement and we prove that the theory of L(Ordω1)L(Ord^{\omega_1}) with parameters in P(ω1)P(\omega_1) is generically invariant for stationary set preserving forcings that preserve MM+++MM^{+++}. Finally we argue that the work of Larson and Asper\'o shows that this is a next to optimal generalization to the Chang model L(Ordω1)L(Ord^{\omega_1}) of Woodin's generic absoluteness results for the Chang model L(Ordω)L(Ord^{\omega}). It remains open whether MM+++MM^{+++} and MM++MM^{++} are equivalent axioms modulo large cardinals and whether MM++MM^{++} suffices to prove the same generic absoluteness results for the Chang model L(Ordω1)L(Ord^{\omega_1}).Comment: - to appear on the Journal of the American Mathemtical Societ

    Cycle decompositions: from graphs to continua

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    We generalise a fundamental graph-theoretical fact, stating that every element of the cycle space of a graph is a sum of edge-disjoint cycles, to arbitrary continua. To achieve this we replace graph cycles by topological circles, and replace the cycle space of a graph by a new homology group for continua which is a quotient of the first singular homology group H1H_1. This homology seems to be particularly apt for studying spaces with infinitely generated H1H_1, e.g. infinite graphs or fractals.Comment: Advances in Mathematics (2011
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