Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-Gutiérrez; Paul J. Szeptycki

Displaying similar documents to “Wijsman hyperspaces of non-separable metric spaces”

On the tangency of sets in generalized metric spaces for certain functions of the class $F_{ρ}^{*}$

T. Konik (1991)

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Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{α}$ , $Y_{α}$ and $Z_{α}$ such that (i) $f X_{α}, f Y_{α}, f Z_{α} = ω ₀$ , where f is either trdef or ₀-trsur, (ii) $A (α) - t r i n d X_{α} = \infty$ and $M (α) - t r i n d X_{α} = - 1$ , (iii) $A (α) - t r i n d Y_{α} = - 1$ and $M (α) - t r i n d Y_{α} = \infty$ , and (iv) $A (α) - t r i n d Z_{α} = M (α) - t r i n d Z_{α} = \infty$ and $A (α + 1) \cap M (α + 1) - t r i n d Z_{α} = - 1$ . We also show that there exists no separable metrizable space $W_{α}$ with $A (α) - t r i n d W_{α} \neq \infty$ , $M (α) - t r i n d W_{α} \neq \infty$ and $A (α) \cap M (α) - t r i n d W_{α} = \infty$ , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

The Lindelöf property in Banach spaces

B. Cascales, I. Namioka, J. Orihuela (2003)

Studia Mathematica

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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent: (i) K is fragmented by $d_{D}$ , where, for each S ⊂ D, $d_{S} (x, y) = s u p ϱ (x (t), y (t)) : t \in S$ . (ii) For each countable subset...

Addition theorems for dense subspaces

Aleksander V. Arhangel'skii (2015)

Commentationes Mathematicae Universitatis Carolinae

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We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$ -space. However, if a normal space $X$ is the union of a finite family $μ$ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable...

Extending generalized Whitney maps

Ivan Lončar (2017)

Archivum Mathematicum

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For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^{X}$ can be extended to a Whitney map for $2^{X}$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.

Wasserstein metric and subordination

Philippe Clément, Wolfgang Desch (2008)

Studia Mathematica

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Let $(X, d_{X})$ , $(Ω, d_{Ω})$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_{p}$ the p-Wasserstein metric with some p ∈ [1,∞), and by $_{p} (X)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f : Ω \to_{p} (X)$ , $ω \mapsto f_{ω}$ , be a Borel map such that f is a contraction from $(Ω, d_{Ω})$ into $(_{p} (X), W_{p})$ . Let ν₁,ν₂ be probability measures on Ω with $W_{p} (ν ₁, ν ₂)$ finite. On X we consider the subordinated measures $μ_{i} = \int_{Ω} f_{ω} d ν_{i} (ω)$ . Then $W_{p} (μ ₁, μ ₂) \leq W_{p} (ν ₁, ν ₂)$ . As an application we show that the solution measures $ϱ_{α} (t)$ ...

Metric unconditionality and Fourier analysis

Stefan Neuwirth (1998)

Studia Mathematica

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We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_{E}^{p} ()$ and $C_{E} ()$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_{E} ()$ , p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_{E}^{p} ()$ ...

General position properties in fiberwise geometric topology

Taras Banakh, Vesko Valov

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General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish $L C^{n - 1}$ -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...

Spaces with property $(D C (ω_{1}))$

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if $X$ is a first countable space with property $(D C (ω_{1}))$ and with a $G_{δ}$ -diagonal then the cardinality of $X$ is at most $𝔠$ . We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$ .

Characterizations of $z$ -Lindelöf spaces

Ahmad Al-Omari, Takashi Noiri (2017)

Archivum Mathematicum

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A topological space $(X, τ)$ is said to be $z$ -Lindelöf [1] if every cover of $X$ by cozero sets of $(X, τ)$ admits a countable subcover. In this paper, we obtain new characterizations and preservation theorems of $z$ -Lindelöf spaces.

Cardinal invariants for κ-box products: weight, density character and Suslin number

W. W. Comfort, Ivan S. Gotchev

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The symbol ${(X_{I})}_{κ}$ (with κ ≥ ω) denotes the space $X_{I} : = \prod_{i \in I} X_{i}$ with the κ-box topology; this has as base all sets of the form $U = \prod_{i \in I} U_{i}$ with $U_{i}$ open in $X_{i}$ and with $| i \in I : U_{i} \neq X_{i} | < κ$ . The symbols w, d and S denote respectively the weight, density character and Suslin number. Generalizing familiar classical results, the authors show inter alia: Theorem 3.1.10(b). If κ ≤ α⁺, |I| = α and each $X_{i}$ contains the discrete space 0,1 and satisfies $w (X_{i}) \leq α$ , then $w (X_{κ}) = α^{< κ}$ . Theorem 4.3.2. If $ω \leq κ \leq | I | \leq 2^{α}$ and $X = {(D (α))}^{I}$ with D(α) discrete, |D(α)| = α, then $d ({(X_{I})}_{κ}) = α^{< κ}$ . Corollaries 5.2.32(a)...

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when $X_{M} = X$ , where $X_{M}$ is the elementary submodel topology on X ∩ M, especially in the case when $X_{M}$ is compact.

On $n$ -thin dense sets in powers of topological spaces

Adam Bartoš (2016)

Commentationes Mathematicae Universitatis Carolinae

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A subset of a product of topological spaces is called $n$ -thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_{3}$ space $X$ without isolated points such that $X^{n}$ contains an $n$ -thin dense subset, but $X^{n + 1}$ does not contain any $n$ -thin dense subset. We also observe that part of the construction can be carried out under MA.

Extension properties of Stone-Čech coronas and proper absolute extensors

A. Chigogidze (2013)

Fundamenta Mathematicae

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We characterize, in terms of X, the extensional dimension of the Stone-Čech corona βX∖X of a locally compact and Lindelöf space X. The non-Lindelöf case is also settled in terms of extending proper maps with values in $I^{τ} ∖ L$ , where L is a finite complex. Further, for a finite complex L, an uncountable cardinal τ and a $Z_{τ}$ -set X in the Tikhonov cube $I^{τ}$ we find a necessary and sufficient condition, in terms of $I^{τ} ∖ X$ , for X to be in the class AE([L]). We also introduce a concept of a proper absolute...

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X, d, μ)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B (x, r)$ converge to $f (x)$ when $r$ converges to $0$ . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

$R_{z}$ -supercontinuous functions

Davinder Singh, Brij Kishore Tyagi, Jeetendra Aggarwal, Jogendra K. Kohli (2015)

Mathematica Bohemica

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A new class of functions called “ $R_{z}$ -supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of $R_{z}$ -supercontinuous functions properly includes the class of $R_{cl}$ -supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of $cl$ -supercontinuous ( $\equiv$ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983),...

On universality of countable and weak products of sigma hereditarily disconnected spaces

Taras Banakh, Robert Cauty (2001)

Fundamenta Mathematicae

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Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power $X^{ω}$ of any subspace X ⊂ Y is not universal for the class ₂ of absolute $G_{δ σ}$ -sets; moreover, if Y is an absolute $F_{σ δ}$ -set, then $X^{ω}$ contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute $G_{δ}$ -set, then $X^{ω}$ contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable...

Functionally countable subalgebras and some properties of the Banaschewski compactification

A. R. Olfati (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a zero-dimensional space and $C_{c} (X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_{c}^{K} (X)$ (resp., $C_{c}^{ψ} (X))$ the set of all functions in $C_{c} (X)$ with compact (resp., pseudocompact) support. First, we observe that $C_{c}^{K} (X) = O_{c}^{β_{0} X ∖ X}$ (resp., $C_{c}^{ψ} (X) = M_{c}^{β_{0} X ∖ υ_{0} X}$ ), where $β_{0} X$ is the Banaschewski compactification of $X$ and $υ_{0} X$ is the $ℕ$ -compactification of $X$ . This implies that for an $ℕ$ -compact space $X$ , the intersection of all free maximal ideals in $C_{c} (X)$ is equal to $C_{c}^{K} (X)$ , i.e., $M_{c}^{β_{0} X ∖ X} = C_{c}^{K} (X)$ . By applying...

On non-normality points, Tychonoff products and Suslin number

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let a space $X$ be Tychonoff product $\prod_{α < τ} X_{α}$ of $τ$ -many Tychonoff nonsingle point spaces $X_{α}$ . Let Suslin number of $X$ be strictly less than the cofinality of $τ$ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $β X$ . In particular, this is true if $X$ is either $R^{τ}$ or $ω^{τ}$ and a cardinal $τ$ is infinite and not countably cofinal.

When $C_{p} (X)$ is domain representable

William Fleissner, Lynne Yengulalp (2013)

Fundamenta Mathematicae

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Let M be a metrizable group. Let G be a dense subgroup of $M^{X}$ . We prove that if G is domain representable, then $G = M^{X}$ . The following corollaries answer open questions. If X is completely regular and $C_{p} (X)$ is domain representable, then X is discrete. If X is zero-dimensional, T₂, and $C_{p} (X,)$ is subcompact, then X is discrete.