Displaying similar documents to “Eberlein spaces of finite metrizability number”

The Dugundji extension property can fail in ωµ -metrizable spaces

Ian Stares, Jerry Vaughan (1996)

Fundamenta Mathematicae

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We show that there exist ω μ -metrizable spaces which do not have the Dugundji extension property ( 2 ω 1 with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.

Gδ -sets in topological spaces and games

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 O n from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset T n of X. The players must obey the rule that O n O n + 1 T n + 1 T n 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 G δ -set. To what extent is the converse true? We show that:  (A) For ℱ the collection of countable subsets of X:   1....

Convexity ranks in higher dimensions

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...

A Ramsey theorem for polyadic spaces

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 ( α κ ) ω is not a universal preimage for uniform Eberlein compact spaces of weight at most κ, thus answering a question of Y. Benyamini, M. Rudin...

If it looks and smells like the reals...

Franklin Tall (2000)

Fundamenta Mathematicae

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Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define X M to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if X M is homeomorphic to ℝ, then X = X M . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.

Each nowhere dense nonvoid closed set in Rn is a σ-limit set

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 , n ≥ 1, is a σ-limit set for some continuous map.

Ordered spaces with special bases

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 G δ -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...

Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke

Hisao Kato (1994)

Fundamenta Mathematicae

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A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that d ( f n ( x ) , f n ( y ) ) > c (resp. d i a m f n ( A ) > c ). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dim X > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate...

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 ω 1 are classified up to forcing equivalence.

Nonseparable Radon measures and small compact spaces

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 [ 0 , 1 ] κ (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 ω 1 null sets in 2 ω 1 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...

Are initially ω 1 -compact separable regular spaces compact?

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.

Strong Fubini properties of ideals

Ireneusz Recław, Piotr Zakrzewski (1999)

Fundamenta Mathematicae

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 Let I and J be σ-ideals on Polish spaces X and Y, respectively. We say that the pair ⟨I,J⟩ has the Strong Fubini Property (SFP) if for every set D ⊆ X× Y with measurable sections, if all its sections D x = y : x , y D are in J, then the sections D y = x : x , y D are in I for every y outside a set from J (“measurable" means being a member of the σ-algebra of Borel sets modulo sets from the respective σ-ideal). We study the question of which pairs of σ-ideals have the Strong Fubini Property. Since CH excludes this...

Standardness of sequences of σ-fields given by certain endomorphisms

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 , E - 1 , E - 2 , . . . A central example is the one-sided shift σ on X = 0 , 1 with 1 2 , 1 2 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 ( x , y ) ( σ ( x ) , T x ( 1 ) ( y ) ) . 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...

Raising dimension under all projections

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 n by looking at its orthogonal projections into hyperplanes?” - we construct a Cantor set in 3 each of whose projections into 2-planes is 1-dimensional. We also consider projections of Cantor sets in n whose images contain open sets, expanding on a result of Borsuk.