A hit-and-miss topology for $2^X$, Cₙ(X) and Fₙ(X)

Benjamín Espinoza; Verónica Martínez-de-la-Vega; Jorge M. Martínez-Montejano

Displaying similar documents to “A hit-and-miss topology for $2^{X}$ , Cₙ(X) and Fₙ(X)”

Locally $G_{δ}$ -spaces

Zdeněk Frolík (1962)

Czechoslovak Mathematical Journal

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Connectedness of some rings of quotients of $C (X)$ with the $m$ -topology

F. Azarpanah, M. Paimann, A. R. Salehi (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this article we define the $m$ -topology on some rings of quotients of $C (X)$ . Using this, we equip the classical ring of quotients $q (X)$ of $C (X)$ with the $m$ -topology and we show that $C (X)$ with the $r$ -topology is in fact a subspace of $q (X)$ with the $m$ -topology. Characterization of the components of rings of quotients of $C (X)$ is given and using this, it turns out that $q (X)$ with the $m$ -topology is connected if and only if $X$ is a pseudocompact almost $P$ -space, if and only if $C (X)$ with $r$ -topology is connected. We also...

Some properties of the topology of $α$ -sets

D. Andrijević (1984)

Matematički Vesnik

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The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology

Katsuro Sakai, Zhongqiang Yang (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $C o n v_{F} (ℝ ⁿ)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $C o n v_{F} (ℝ ⁿ) \approx ℝ ⁿ \times Q$ for every n > 1 whereas $C o n v_{F} (ℝ) \approx ℝ \times$ .

The AR-Property of the spaces of closed convex sets

Katsuro Sakai, Masato Yaguchi (2006)

Colloquium Mathematicae

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Let $C o n v_{H} (X)$ , $C o n v_{A W} (X)$ and $C o n v_{W} (X)$ be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of $C o n v_{H} (X)$ and the space $C o n v_{A W} (X)$ are AR. In case X is separable, $C o n v_{W} (X)$ is locally path-connected.

A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X, τ_{X})$ and $(Y, τ_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_{X}$ and $f^{- 1} (U) : U \in τ_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact...

$O^{*}$ -Topologies on the Test Function Algebra

G. Lassner (1975)

Publications du Département de mathématiques (Lyon)

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The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

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Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k} (X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_{k} (X)$ is Ascoli iff $C_{k} (X)$ is a $k_{ℝ}$ -space iff X is locally compact. Moreover, $C_{k} (X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

On two-to-one continuous functions

J. Mioduszewski

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CONTENTSIntroduction................................................................................................................................................................................3I. General properties of k-to-one functions on locally compact spaces1. Multi-valued functions Ф and ψ......................................................................................................................................... 62. The proof of (I.11)..................................................................................................................................................................

On affinity of Peano type functions

Tomasz Słonka (2012)

Colloquium Mathematicae

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We show that if n is a positive integer and $2^{ℵ ₀} \leq ℵ ₙ$ , then for every positive integer m and for every real constant c > 0 there are functions $f ₁, . . ., f_{n + m} : ℝ ⁿ \to ℝ$ such that $(f ₁, . . ., f_{n + m}) (ℝ ⁿ) = ℝ^{n + m}$ and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that $(f_{i ₁}, . . ., f_{i ₙ}) (y) = y + w$ for $y \in x + (- c, c) \times ℝ^{n - 1}$ .

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.

A continuum $X$ such that $C (X)$ is not continuously homogeneous

Alejandro Illanes (2016)

Commentationes Mathematicae Universitatis Carolinae

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A metric continuum $X$ is said to be continuously homogeneous provided that for every two points $p, q \in X$ there exists a continuous surjective function $f : X \to X$ such that $f (p) = q$ . Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum $X$ such that the hyperspace of subcontinua of $X$ , $C (X)$ , is not continuously homogeneous.

Kempisty's theorem for the integral product quasicontinuity

Zbigniew Grande (2006)

Colloquium Mathematicae

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A function f: ℝⁿ → ℝ satisfies the condition $Q_{i} (x)$ (resp. $Q_{s} (x)$ , $Q_{o} (x)$ ) at a point x if for each real r > 0 and for each set U ∋ x open in the Euclidean topology of ℝⁿ (resp. strong density topology, ordinary density topology) there is an open set I such that I ∩ U ≠ ∅ and $| (1 / μ (U \cap I)) \int_{U \cap I} f (t) d t - f (x) | < r$ . Kempisty’s theorem concerning the product quasicontinuity is investigated for the above notions.

Topological properties of some spaces of continuous operators

Marian Nowak (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_{b} (X, E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space $L_{β} (C_{b} (X, E), F)$ of all $(β, | | \cdot | |_{F})$ -continuous linear operators from $C_{b} (X, E)$ to F, equipped with the topology $τ_{s}$ of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize $τ_{s}$ -compact subsets of $L_{β} (C_{b} (X, E), F)$ in terms of properties of the corresponding sets of the representing...

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)...

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...

Domain representability of $C_{p} (X)$

Harold Bennett, David Lutzer (2008)

Fundamenta Mathematicae

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Let $C_{p} (X)$ be the space of continuous real-valued functions on X, with the topology of pointwise convergence. We consider the following three properties of a space X: (a) $C_{p} (X)$ is Scott-domain representable; (b) $C_{p} (X)$ is domain representable; (c) X is discrete. We show that those three properties are mutually equivalent in any normal T₁-space, and that properties (a) and (c) are equivalent in any completely regular pseudo-normal space. For normal spaces, this generalizes the recent result of Tkachuk...

An irrational problem

Franklin D. Tall (2002)

Fundamenta Mathematicae

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Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define $X_{M}$ to be X ∩ M with topology generated by $U \cap M : U \in \cap M$ . Suppose $X_{M}$ is homeomorphic to the irrationals; must $X = X_{M}$ ? We have partial results. We also answer a question of Gruenhage by showing that if $X_{M}$ is homeomorphic to the “Long Cantor Set”, then $X = X_{M}$ .

Displaying similar documents to “A hit-and-miss topology for 2 X , Cₙ(X) and Fₙ(X)”

Displaying similar documents to “A hit-and-miss topology for $2^{X}$ , Cₙ(X) and Fₙ(X)”