Generalized Helly spaces, continuity of monotone functions, and metrizing maps

Lech Drewnowski; Artur Michalak

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Displaying similar documents to “Generalized Helly spaces, continuity of monotone functions, and metrizing maps”

Modifications of the double arrow space and related Banach spaces C(K)

Witold Marciszewski (2008)

Studia Mathematica

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We consider the class of compact spaces $K_{A}$ which are modifications of the well known double arrow space. The space $K_{A}$ is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of $K_{A}$ spaces and on the isomorphic classification of the Banach spaces $C (K_{A})$ .

The positive cone of a Banach lattice. Coincidence of topologies and metrizability

Zbigniew Lipecki (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a Banach lattice, and denote by $X_{+}$ its positive cone. The weak topology on $X_{+}$ is metrizable if and only if it coincides with the strong topology if and only if $X$ is Banach-lattice isomorphic to $l^{1} (Γ)$ for a set $Γ$ . The weak $^{*}$ topology on $X_{+}^{*}$ is metrizable if and only if $X$ is Banach-lattice isomorphic to a $C (K)$ -space, where $K$ is a metrizable compact space.

Spaces of continuous step functions over LOTS

Raushan Z. Buzyakova (2006)

Fundamenta Mathematicae

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We investigate spaces $C_{p} (\cdot, n)$ over LOTS (linearly ordered topological spaces). We find natural necessary conditions for linear Lindelöfness of $C_{p} (\cdot, n)$ over LOTS. We also characterize countably compact LOTS whose $C_{p} (\cdot, n)$ is linearly Lindelöf for each n. Both the necessary conditions and the characterization are given in terms of the topology of the Dedekind completion of a LOTS.

True preimages of compact or separable sets for functional analysts

Lech Drewnowski (2020)

Commentationes Mathematicae Universitatis Carolinae

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We discuss various results on the existence of ‘true’ preimages under continuous open maps between $F$ -spaces, $F$ -lattices and some other spaces. The aim of the paper is to provide accessible proofs of this sort of results for functional-analysts.

Isometric embeddings of a class of separable metric spaces into Banach spaces

Sophocles K. Mercourakis, Vassiliadis G. Vassiliadis (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $(M, d)$ be a bounded countable metric space and $c > 0$ a constant, such that $d (x, y) + d (y, z) - d (x, z) \geq c$ , for any pairwise distinct points $x, y, z$ of $M$ . For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $ℓ_{\infty}$ .

Counting linearly ordered spaces

Gerald Kuba (2014)

Colloquium Mathematicae

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For a transfinite cardinal κ and i ∈ 0,1,2 let $ℒ_{i} (κ)$ be the class of all linearly ordered spaces X of size κ such that X is totally disconnected when i = 0, the topology of X is generated by a dense linear ordering of X when i = 1, and X is compact when i = 2. Thus every space in ℒ₁(κ) ∩ ℒ₂(κ) is connected and hence ℒ₁(κ) ∩ ℒ₂(κ) = ∅ if $κ < 2^{ℵ ₀}$ , and ℒ₀(κ) ∩ ℒ₁(κ) ∩ ℒ₂(κ) = ∅ for arbitrary κ. All spaces in ℒ₁(ℵ₀) are homeomorphic, while ℒ₂(ℵ₀) contains precisely ℵ₁ spaces up to homeomorphism. The...

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

T. Konik (1991)

Matematički Vesnik

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On Surjective Bing Maps

Hisao Kato, Eiichi Matsuhashi (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense $G_{δ}$ -subset of the space of all maps. In [6], J. Krasinkiewicz independently proved that the set of all Bing maps of a compact metric space to an n-dimensional manifold (n ≥ 1) is a dense $G_{δ}$ -subset of the space of maps. In [9], J. Song and E. D. Tymchatyn, solving some problems of J. Krasinkiewicz ([6]), proved that the set of all Bing maps of a compact metric space to a nondegenerate...

Chromatic number of the product of graphs, graph homomorphisms, antichains and cofinal subsets of posets without AC

Amitayu Banerjee, Zalán Gyenis (2021)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. $\circ$ If in a partially ordered set, all chains are finite and all antichains have size $ℵ_{α}$ , then the set has size $ℵ_{α}$ for any regular $ℵ_{α}$ . $\circ$ Every partially ordered set without a maximal element has two disjoint cofinal sub sets – CS. $\circ$ Every partially...

Well-posedness of optimization problems and Hausdorff metric on partial maps

Alessandro Caterino, Rita Ceppitelli, Ľubica Holà (2006)

Bollettino dell'Unione Matematica Italiana

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The object of this paper is the Hausdorff metric topology on partial maps with closed domains. This topological space is denoted by $(\mathcal{P},H_{\rho})$ . An equivalence of well-posedness of constrained continuous problems is proved. By using the completeness of the Hausdorff metric on the space of usco maps with moving domains, the complete metrizability of $(\mathcal{P},H_{\rho})$ is investigated.

Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

Dale E. Alspach, Elói Medina Galego (2011)

Studia Mathematica

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A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C (ω^{ω})$ then X contains a copy...

The nonexistence of universal metric flows

Stefan Geschke (2018)

Commentationes Mathematicae Universitatis Carolinae

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We consider dynamical systems of the form $(X, f)$ where $X$ is a compact metric space and $f : X \to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $ω$ -limit sets, answering a question by Will Brian.

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.

Some remarks providing discontinuous maps on some $C_{p} (X)$ spaces

S. Moll (2008)

Banach Center Publications

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Let X be a completely regular Hausdorff topological space and $C_{p} (X)$ the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space $C_{p} (X)$ , if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of $C_{p} (X)$ .

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f : {[0, 1]}^{m} \to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f : {[0, 1]}^{m} \to X$ with respect to any norming subset there exists a separately increasing function $g : {[0, 1]}^{m} \to ℝ$ such that the sets of...

Hyperreflexivity of bilattices

Kamila Kliś-Garlicka (2016)

Czechoslovak Mathematical Journal

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The notion of a bilattice was introduced by Shulman. A bilattice is a subspace analogue for a lattice. In this work the definition of hyperreflexivity for bilattices is given and studied. We give some general results concerning this notion. To a given lattice $ℒ$ we can construct the bilattice $Σ_{ℒ}$ . Similarly, having a bilattice $Σ$ we may consider the lattice $ℒ_{Σ}$ . In this paper we study the relationship between hyperreflexivity of subspace lattices and of their associated bilattices. Some examples...

A class of multiplicative lattices

Tiberiu Dumitrescu, Mihai Epure (2021)

Czechoslovak Mathematical Journal

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We study the multiplicative lattices $L$ which satisfy the condition $a = (a : (a : b)) (a : b)$ for all $a, b \in L$ . Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group $ℤ$ or $ℝ$ . A sharp lattice $L$ localized at its maximal elements are totally ordered sharp lattices. The converse is true if $L$ has finite character.

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...