A classification of definable forcings on ω1
Jindřich Zapletal (1997)
Fundamenta Mathematicae
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Under the assumption of the existence of sharps for reals all simply definable posets on are classified up to forcing equivalence.
Jindřich Zapletal (1997)
Fundamenta Mathematicae
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Under the assumption of the existence of sharps for reals all simply definable posets on are classified up to forcing equivalence.
Franklin Tall (2000)
Fundamenta Mathematicae
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Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if is homeomorphic to ℝ, then . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.
Jacob Feldman, Daniel Rudolph (1998)
Fundamenta Mathematicae
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Let E be an ergodic endomorphism of the Lebesgue probability space X, ℱ, μ. It gives rise to a decreasing sequence of σ-fields A central example is the one-sided shift σ on with product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphismis defined on (X× Y, μ× ν) by . Here ℱ is the σ-field of μ× ν-measurable sets. Each field is a two-point extension of the one beneath it. Vershik has defined as “standard” any decreasing sequence of...
Andrei Sivak (1996)
Fundamenta Mathematicae
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We discuss main properties of the dynamics on minimal attraction centers (σ-limit sets) of single trajectories for continuous maps of a compact metric space into itself. We prove that each nowhere dense nonvoid closed set in , n ≥ 1, is a σ-limit set for some continuous map.
Winfried Just, Marion Scheepers, Juris Steprans, Paul Szeptycki (1997)
Fundamenta Mathematicae
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Players ONE and TWO play the following game: In the nth inning ONE chooses a set from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset of X. The players must obey the rule that for each n. TWO wins if the intersection of TWO’s sets is equal to the union of ONE’s sets. If ONE has no winning strategy, then each element of ℱ is a -set. To what extent is the converse true? We show that: (A) For ℱ the collection of countable subsets of X: 1....
Menachem Kojman (2000)
Fundamenta Mathematicae
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A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a...
Robert Kallman (2000)
Fundamenta Mathematicae
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Every reasonably sized matrix group has an injective homomorphism into the group of all bijections of the natural numbers. However, not every reasonably sized simple group has an injective homomorphism into .
Alan Dow, Istvan Juhász (1997)
Fundamenta Mathematicae
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We investigate the question of the title. While it is immediate that CH yields a positive answer we discover that the situation under the negation of CH holds some surprises.
Grzegorz Plebanek (1997)
Fundamenta Mathematicae
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We investigate the problem if every compact space K carrying a Radon measure of Maharam type κ can be continuously mapped onto the Tikhonov cube (κ being an uncountable cardinal). We show that for κ ≥ cf(κ) ≥ κ this holds if and only if κ is a precaliber of measure algebras. Assuming that there is a family of null sets in such that every perfect set meets one of them, we construct a compact space showing that the answer to the above problem is “no” for κ = ω. We also give alternative...
Kenneth A. Ribet (1997)
Acta Arithmetica
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We discuss the equation in which a, b, and c are non-zero relatively prime integers, p is an odd prime number, and α is a positive integer. The technique used to prove Fermat’s Last Theorem shows that the equation has no solutions with α < 1 or b even. When α=1 and b is odd, there are the two trivial solutions (±1, ∓ 1, ±1). In 1952, Dénes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for p≡ 1 mod 4.
Sergiĭ Kolyada, Michał Misiurewicz, L’ubomír Snoha (1999)
Fundamenta Mathematicae
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The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces and a sequence of continuous maps , , is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of . As an application we construct a large class of smooth triangular maps of the square of type and...
Wiesław Główczyński, Bohuslav Balcar, Thomas Jech (1998)
Fundamenta Mathematicae
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We investigate the sequential topology on a complete Boolean algebra B determined by algebraically convergent sequences in B. We show the role of weak distributivity of B in separation axioms for the sequential topology. The main result is that a necessary and sufficient condition for B to carry a strictly positive Maharam submeasure is that B is ccc and that the space is Hausdorff. We also characterize sequential cardinals.
John Cobb (1994)
Fundamenta Mathematicae
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As a special case of the general question - “What information can be obtained about the dimension of a subset of by looking at its orthogonal projections into hyperplanes?” - we construct a Cantor set in each of whose projections into 2-planes is 1-dimensional. We also consider projections of Cantor sets in whose images contain open sets, expanding on a result of Borsuk.
A. Katok, A. Mezhirov (1998)
Fundamenta Mathematicae
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Let f be a continuous map of the circle or the interval I into itself, piecewise , piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least periodic points of period with large derivative along the period, for some subsequence of natural numbers. For a strictly monotone map f without critical points we show the existence of at least such points.
Sebastian van Strien (2000)
Fundamenta Mathematicae
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We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if is critically finite with non-degenerate critical point such that are hyperbolic periodic points for i = 1,...,n, then IV-1. Age impartible......................................................................................................................................................................... 31 is a local diffeomorphism...
Murray Bell (1996)
Fundamenta Mathematicae
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A polyadic space is a Hausdorff continuous image of some power of the one-point compactification of a discrete space. We prove a Ramsey-like property for polyadic spaces which for Boolean spaces can be stated as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint. One corollary is that is not a universal preimage for uniform Eberlein compact spaces of weight at most κ, thus answering a question of Y. Benyamini, M. Rudin...
Harold Bennett, David Lutzer (1998)
Fundamenta Mathematicae
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We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a -diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized...
K. Győry (1997)
Acta Arithmetica
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P. 294, line 14: For “Satz 8” read “Satz 7”, and for “equation (10)” read “equation (13)”.