Some remarks providing discontinuous maps on some $C_p(X)$ spaces

S. Moll

Displaying similar documents to “Some remarks providing discontinuous maps on some $C_{p} (X)$ spaces”

On open maps and related functions over the Salbany compactification

Mbekezeli Nxumalo (2024)

Archivum Mathematicum

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Given a topological space $X$ , let $𝒰 X$ and $η_{X} : X \to 𝒰 X$ denote, respectively, the Salbany compactification of $X$ and the compactification map called the Salbany map of $X$ . For every continuous function $f : X \to Y$ , there is a continuous function $𝒰 f : 𝒰 X \to 𝒰 Y$ , called the Salbany lift of $f$ , satisfying $(𝒰 f) \circ η_{X} = η_{Y} \circ f$ . If a continuous function $f : X \to Y$ has a stably compact codomain $Y$ , then there is a Salbany extension $F : 𝒰 X \to Y$ of $f$ , not necessarily unique, such that $F \circ η_{X} = f$ . In this paper, we give a condition on a space such that its Salbany map is open. In...

Finite-dimensional maps and dendrites with dense sets of end points

Hisao Kato, Eiichi Matsuhashi (2006)

Colloquium Mathematicae

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The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space $C (X, I^{p + 2 k + 1 - i})$ such that the diagonal product $f \times g : X \to Y \times I^{p + 2 k + 1 - i}$ is an (i+1)-to-1 map is a dense $G_{δ}$ -subset of $C (X, I^{p + 2 k + 1 - i})$ . In this paper, we prove that if f: X → Y is as above and $D_{j}$ (j = 1,..., k) are superdendrites, then the set of maps h in $C (X, \prod_{j = 1}^{k} D_{j} \times I^{p + 1 - i})$ such that $f \times h : X \to Y \times (\prod_{j = 1}^{k} D_{j} \times I^{p + 1 - i})$ is (i+1)-to-1 is a dense $G_{δ}$ -subset of $C (X, \prod_{j = 1}^{k} D_{j} \times I^{p + 1 - i})$ for each 0 ≤ i ≤ p.

Assertion Q distinguishes topologically ω* and $𝔪$ when $𝔪$ regular and $𝔪 > ω$

R. Frankiewicz (1978)

Colloquium Mathematicae

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$R_{z}$ -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),...

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ} (X)$ ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ} (Y)$ ), and $ϕ_{\infty}$ a linear isometry from M into C(Y) (resp., $C_{ℝ} (Y)$ ). We show, under the assumption that $Π_{N} \subset Π_{T}$ , where $Π_{N}$ is...

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...

A new function space and applications

Jean Bourgain, Haïm Brezis, Petru Mironescu (2015)

Journal of the European Mathematical Society

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We define a new function space $B$ , which contains in particular BMO, BV, and $W^{1 / p, p}$ , $1 < p < \infty$ . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^{p}$ norms of integer-valued functions in $B$ . We introduce a significant closed subspace, $B_{0}$ , of $B$ , containing in particular VMO and $W^{1 / p, p}$ , $1 \leq p < \infty$ . The above mentioned estimates imply in particular that integer-valued functions belonging to $B_{0}$ are necessarily constant. This framework provides a “common roof”...

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials $P_{c} (z) = z^{d} + c$ for c ∈ ℂ. Denote by $J_{c}$ the Julia set of $P_{c}$ and let $ℳ_{d} = c | J_{c} i s c o n n e c t e d$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c ₀ \in \partial ℳ_{d}$ : those for which the critical point 0 is not recurrent by $P_{c ₀}$ and without parabolic cycles. The Hausdorff dimension of $J_{c}$ , denoted by $H D (J_{c})$ , does not depend continuously on c at such $c ₀ \in \partial ℳ_{d}$ ; on the other hand the function $c \mapsto H D (J_{c})$ is analytic in $ℂ - ℳ_{d}$ . Our first result asserts that there is still some...

Compactness in $C_{s} (T)$ and Applications

Richard Haydon (1972)

Publications du Département de mathématiques (Lyon)

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A compactness result for polyharmonic maps in the critical dimension

Shenzhou Zheng (2016)

Czechoslovak Mathematical Journal

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For $n = 2 m \geq 4$ , let $Ω \in ℝ^{n}$ be a bounded smooth domain and $𝒩 \subset ℝ^{L}$ a compact smooth Riemannian manifold without boundary. Suppose that ${u_{k}} \in W^{m, 2} (Ω, 𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$ -polyharmonic maps $\frac{d}{d t} |_{t = 0} E_{m} (Π (u + t ξ)) = 0$ with $Φ_{k} \to 0$ in ${(W^{m, 2} (Ω, 𝒩))}^{*}$ and $u_{k} ⇀ u$ weakly in $W^{m, 2} (Ω, 𝒩)$ . Then $u$ is an $m$ -polyharmonic map. In particular, the space of $m$ -polyharmonic maps is sequentially compact for the weak- $W^{m, 2}$ topology.

A localization property for $B_{p q}^{s}$ and $F_{p q}^{s}$ spaces

Hans Triebel (1994)

Studia Mathematica

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Let $f^{j} = \sum_{k} a_{k} f (2^{j + 1} x - 2 k)$ , where the sum is taken over the lattice of all points k in $ℝ^{n}$ having integer-valued components, j∈ℕ and $a_{k} \in ℂ$ . Let $A_{p q}^{s}$ be either $B_{p q}^{s}$ or $F_{p q}^{s}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^{n} .$ The aim of the paper is to clarify under what conditions $∥ f^{j} | A_{p q}^{s} ∥$ is equivalent to $2^{j (s - n / p)} (\sum_{k} | a_{k} |^{p})^{1 / p} ∥ f | A_{p q}^{s} ∥$ .

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane Baladi, Daniel Smania (2012)

Annales scientifiques de l'École Normale Supérieure

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We consider $C^{2}$ families $t \mapsto f_{t}$ of $C^{4}$ unimodal maps $f_{t}$ whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure $μ_{t}$ of $f_{t}$ depends differentiably on $t$ , as a distribution of order $1$ . The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of $μ_{t}$ for a Benedicks-Carleson map $f_{t}$ , in terms of a single smooth function and the...

$α$ -continuous multivalued maps

J. Eweret (1989)

Matematički Vesnik

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Interaction between cellularity of Alexandroff spaces and entropy of generalized shift maps

Fatemah Ayatollah Zadeh Shirazi, Sahar Karimzadeh Dolatabad, Sara Shamloo (2016)

Commentationes Mathematicae Universitatis Carolinae

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In the following text for a discrete finite nonempty set $K$ and a self-map $ϕ : X \to X$ we investigate interaction between different entropies of generalized shift $σ_{ϕ} : K^{X} \to K^{X}$ , ${(x_{α})}_{α \in X} \mapsto {(x_{ϕ (α)})}_{α \in X}$ and cellularities of some Alexandroff topologies on $X$ .

$Σ_{s}$ -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any $Σ_{s}$ -product of at most $𝔠$ -many $L Σ (\leq ω)$ -spaces has the $L Σ (\leq ω)$ -property. This result generalizes some known results about $L Σ (\leq ω)$ -spaces. On the other hand, we prove that every $Σ_{s}$ -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every $Σ_{s}$ -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

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.

$L_{p, q}$ spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p, q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p, q} (B)$ ........ 235. Examples in $L_{p, q}$ theory...................................................................................

Pluriharmonic extension in proper image domains

Rafał Czyż (2009)

Annales Polonici Mathematici

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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s ≥ 3. Also let $Ω_{π}$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f : \partial Ω_{π} \to ℝ$ that can be extended to a real-valued pluriharmonic function in $Ω_{π}$ .

Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets $J_{0, σ}$ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0, σ}$ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS $ϕ_{σ, α}^{n, k}$ is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

Selivanovski hard sets are hard

Janusz Pawlikowski (2015)

Fundamenta Mathematicae

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Let $H \subseteq Z \subseteq 2^{ω}$ . For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{ω}$ to Z give as preimages of H all Σₙ¹ subsets of $2^{ω}$ , then so do continuous injections.

On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets

Ludwik Jaksztas (2011)

Fundamenta Mathematicae

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Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map $g_{σ}$ . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_{0, σ}$ is continuous at σ₀ as the function of the parameter $σ \in \bar{₀}$ if and only if $H D (J_{0, σ ₀}) \geq 4 / 3$ . Since $H D (J_{0, σ}) > 4 / 3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $H D (J_{0, σ})$ on an open and dense subset of...

Displaying similar documents to “Some remarks providing discontinuous maps on some C p ( X ) spaces”

Displaying similar documents to “Some remarks providing discontinuous maps on some $C_{p} (X)$ spaces”