On D-dimension of metrizable spaces

Wojciech Olszewski

Displaying similar documents to “On D-dimension of metrizable spaces”

Eberlein spaces of finite metrizability number

István Juhász, Zoltán Szentmiklóssy, Andrzej Szymański (2007)

Commentationes Mathematicae Universitatis Carolinae

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Yakovlev [, Comment. Math. Univ. Carolin. (1980), 263–283] showed that any Eberlein compactum is hereditarily $σ$ -metacompact. We show that this property actually characterizes Eberlein compacta among compact spaces of finite metrizability number. Uniformly Eberlein compacta and Corson compacta of finite metrizability number can be characterized in an analogous way.

Forcing countable networks for spaces satisfying $R (X^{ω}) = ω$

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (1996)

Commentationes Mathematicae Universitatis Carolinae

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We show that all finite powers of a Hausdorff space $X$ do not contain uncountable weakly separated subspaces iff there is a c.c.c poset $P$ such that in $V^{P}$ $X$ is a countable union of $0$ -dimensional subspaces of countable weight. We also show that this theorem is sharp in two different senses: (i) we cannot get rid of using generic extensions, (ii) we have to consider all finite powers of $X$ .

A contribution to the topological classification of the spaces Ср(X)

Robert Cauty, Tadeusz Dobrowolski, Witold Marciszewski (1993)

Fundamenta Mathematicae

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We prove that for each countably infinite, regular space X such that $C_{p} (X)$ is a $Z_{σ}$ -space, the topology of $C_{p} (X)$ is determined by the class $F_{0} (C_{p} (X))$ of spaces embeddable onto closed subsets of $C_{p} (X)$ . We show that $C_{p} (X)$ , whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set $Ω_{α}$ for the multiplicative Borel class $M_{α}$ if $F_{0} (C_{p} (X)) = M_{α}$ . For each ordinal α ≥ 2, we provide an example $X_{α}$ such that $C_{p} (X_{α})$ is homeomorphic to $Ω_{α}$ .

Bing maps and finite-dimensional maps

Michael Levin (1996)

Fundamenta Mathematicae

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Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map $g : X \to 𝕀^{k}$ such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open. Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map $g : X \to 𝕀^{k}$ such that dim (f × g) = 1. We improve this result of Sternfeld...

The dimension of X^n where X is a separable metric space

John Kulesza (1996)

Fundamenta Mathematicae

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For a separable metric space X, we consider possibilities for the sequence $S (X) = d_{n} : n \in ℕ$ where $d_{n} = d i m X^{n}$ . In Section 1, a general method for producing examples is given which can be used to realize many of the possible sequences. For example, there is $X_{n}$ such that $S (X_{n}) = n, n + 1, n + 2, . . .$ , $Y_{n}$ , for n >1, such that $S (Y_{n}) = n, n + 1, n + 2, n + 2, n + 2, . . .$ , and Z such that S(Z) = 4, 4, 6, 6, 7, 8, 9,.... In Section 2, a subset X of $ℝ^{2}$ is shown to exist which satisfies $1 = d i m X = d i m X^{2}$ and $d i m X^{3} = 2$ .

On convolution operators with small support which are far from being convolution by a bounded measure

Edmond Granirer (1994)

Colloquium Mathematicae

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Let $C V_{p} (F)$ be the left convolution operators on $L^{p} (G)$ with support included in F and $M_{p} (F)$ denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that $C V_{p} (F)$ , $C V_{p} (F) / M_{p} (F)$ and $C V_{p} (F) / W$ are as big as they can be, namely have $l^{\infty}$ as a quotient, where the ergodic space W contains, and at times is very big relative to $M_{p} (F)$ . Other subspaces of $C V_{p} (F)$ are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.

$Σ$ -products of paracompact Čech-scattered spaces

Hidenori Tanaka (2006)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we shall discuss $Σ$ -products of paracompact Čech-scattered spaces and show the following: (1) Let $Σ$ be a $Σ$ -product of paracompact Čech-scattered spaces. If $Σ$ has countable tightness, then it is collectionwise normal. (2) If $Σ$ is a $Σ$ -product of first countable, paracompact (subparacompact) Čech-scattered spaces, then it is shrinking (subshrinking).

Parametrized Cichoń's diagram and small sets

Janusz Pawlikowski, Ireneusz Recław (1995)

Fundamenta Mathematicae

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We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^{w} \times 2^{w}$ and continuous functions $e, f : w^{w} \to w^{w}$ such that • N is $G_{δ}$ and $N_{x} : x \in w^{w}$ , the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^{w}$ ; • M is $F_{σ}$ and $M_{x} : x \in w^{w}$ is a basis for the ideal of meager subsets of $2^{w}$ ; • $\forall x, y N_{e (x)} \subseteq N_{y} \Rightarrow M_{x} \subseteq M_{f (y)}$ . From this we derive that for a separable metric space X, •if for all Borel (resp. $G_{δ}$ ) sets...

On a discrete version of the antipodal theorem

Krzysztof Oleszkiewicz (1996)

Fundamenta Mathematicae

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The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping $f : S^{k} \to ℝ^{k}$ there exists a point $x \in S^{k}$ such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which $S^{k}$ is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of $i n f_{x} | | f (x) - f (- x) | |$ which were previously known (as far as the author knows) only for f linear (cf. [1]).