Open maps do not preserve Whyburn property

Franco Obersnel

Displaying similar documents to “Open maps do not preserve Whyburn property”

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty} : L^{\infty} (X) \to L ¹ (X)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T \circ i_{\infty} : L^{\infty} (X) \to Y$ is a weakly compact operator. Moreover, we obtain that...

Cellularity and the index of narrowness in topological groups

Mihail G. Tkachenko (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study relations between the cellularity and index of narrowness in topological groups and their $G_{δ}$ -modifications. We show, in particular, that the inequalities $in ({(H)}_{τ}) \leq 2^{τ \cdot in (H)}$ and $c ({(H)}_{τ}) \leq 2^{2^{τ \cdot in (H)}}$ hold for every topological group $H$ and every cardinal $τ \geq ω$ , where ${(H)}_{τ}$ denotes the underlying group $H$ endowed with the $G_{τ}$ -modification of the original topology of $H$ and $in (H)$ is the index of narrowness of the group $H$ . Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $ω$ -narrow group...

A characterization of almost continuity and weak continuity

Chrisostomos Petalas, Theodoros Vidalis (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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It is well known that a function $f$ from a space $X$ into a space $Y$ is continuous if and only if, for every set $K$ in $X$ the image of the closure of $K$ under $f$ is a subset of the closure of the image of it. In this paper we characterize almost continuity and weak continuity by proving similar relations for the subsets $K$ of $X$ .

On projectional skeletons in Vašák spaces

Ondřej F. K. Kalenda (2017)

Commentationes Mathematicae Universitatis Carolinae

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We provide an alternative proof of the theorem saying that any Vašák (or, weakly countably determined) Banach space admits a full $1$ -projectional skeleton. The proof is done with the use of the method of elementary submodels and is comparably simple as the proof given by W. Kubiś (2009) in case of weakly compactly generated spaces.

Weakly precompact subsets of L₁(μ,X)

Ioana Ghenciu (2012)

Colloquium Mathematicae

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Let (Ω,Σ,μ) be a probability space, X a Banach space, and L₁(μ,X) the Banach space of Bochner integrable functions f:Ω → X. Let W = f ∈ L₁(μ,X): for a.e. ω ∈ Ω, ||f(ω)|| ≤ 1. In this paper we characterize the weakly precompact subsets of L₁(μ,X). We prove that a bounded subset A of L₁(μ,X) is weakly precompact if and only if A is uniformly integrable and for any sequence (fₙ) in A, there exists a sequence (gₙ) with $g ₙ \in c o f_{i} : i \geq n$ for each n such that for a.e. ω ∈ Ω, the sequence (gₙ(ω)) is weakly...

Extension and lifting of weakly continuous polynomials

Raffaella Cilia, Joaquín M. Gutiérrez (2005)

Studia Mathematica

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We show that a Banach space X is an ℒ₁-space (respectively, an $ℒ_{\infty}$ -space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space $_{w b} (^{m} X)$ of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an $ℒ_{\infty}$ -space.

On AP spaces in concern with compact-like sets and submaximality

Mi Ae Moon, Myung Hyun Cho, Junhui Kim (2011)

Commentationes Mathematicae Universitatis Carolinae

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The definitions of AP and WAP were originated in categorical topology by A. Pultr and A. Tozzi, Equationally closed subframes and representation of quotient spaces, Cahiers Topologie Géom. Différentielle Catég. 34 (1993), no. 3, 167-183. In general, we have the implications: $T_{2} \Rightarrow K C \Rightarrow U S \Rightarrow T_{1}$ , where $K C$ is defined as the property that every compact subset is closed and $U S$ is defined as the property that every convergent sequence has at most one limit. And a space is called submaximal if every dense subset...

Weakly precompact operators on $C_{b} (X, E)$ with the strict topology

Juliusz Stochmal (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 weakly precompact operators $T : C_{b} (X, E) \to F$ . In particular, we show that if X is a paracompact k-space and E contains no isomorphic copy of l¹, then every strongly bounded operator $T : C_{b} (X, E) \to F$ is weakly precompact.

On weakly Gibson $F_{σ}$ -measurable mappings

Olena Karlova, Volodymyr Mykhaylyuk (2013)

Colloquium Mathematicae

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A function f: X → Y between topological spaces is said to be a weakly Gibson function if $f (Ū) \subseteq \bar{f (U)}$ for any open connected set U ⊆ X. We prove that if X is a locally connected hereditarily Baire space and Y is a T₁-space then an $F_{σ}$ -measurable mapping f: X → Y is weakly Gibson if and only if for any connected set C ⊆ X with dense connected interior the image f(C) is connected. Moreover, we show that each weakly Gibson $F_{σ}$ -measurable mapping f: ℝⁿ → Y, where Y is a T₁-space, has a connected graph. ...

Completely Continuous operators

Ioana Ghenciu, Paul Lewis (2012)

Colloquium Mathematicae

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A Banach space X has the Dunford-Pettis property (DPP) provided that every weakly compact operator T from X to any Banach space Y is completely continuous (or a Dunford-Pettis operator). It is known that X has the DPP if and only if every weakly null sequence in X is a Dunford-Pettis subset of X. In this paper we give equivalent characterizations of Banach spaces X such that every weakly Cauchy sequence in X is a limited subset of X. We prove that every operator T: X → c₀ is completely...

On the size of quotients of function spaces on a topological group

Ahmed Bouziad, Mahmoud Filali (2011)

Studia Mathematica

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For a non-precompact topological group G, we consider the space C(G) of bounded, continuous, scalar-valued functions on G with the supremum norm, together with the subspace LMC(G) of left multiplicatively continuous functions, the subspace LUC(G) of left norm continuous functions, and the subspace WAP(G) of weakly almost periodic functions. We establish that the quotient space LUC(G)/WAP(G) contains a linear isometric copy of $ℓ_{\infty}$ , and that the quotient space C(G)/LMC(G) (and a fortiori...

The point of continuity property, neighbourhood assignments and filter convergences

Ahmed Bouziad (2012)

Fundamenta Mathematicae

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We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment ${(V_{x})}_{x \in X}$ of X such that d(f(x),f(y)) < ε whenever $(x, y) \in V_{y} \times V_{x}$ . We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric...

Weak precompactness and property (V*) in spaces of compact operators

Ioana Ghenciu (2015)

Colloquium Mathematicae

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We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w *} (E *, F)$ (resp. $K_{w *} (E *, F)$ ) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w *} (E *, F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w *} (E *, F)$ has property (wV*). Suppose...