Function space topologies deriving from hypertopologies and networks

A. Di Concilio; A. Miranda

Displaying similar documents to “Function space topologies deriving from hypertopologies and networks”

Note on the Wijsman hyperspaces of completely metrizable spaces

J. Chaber, R. Pol (2002)

Bollettino dell'Unione Matematica Italiana

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We consider the hyperspace $C L (X)$ of nonempty closed subsets of completely metrizable space $X$ endowed with the Wijsman topologies $τ_{W_{d}}$ . If $X$ is separable and $d$ , $e$ are two metrics generating the topology of $X$ , every countable set closed in $(C L (X), τ_{W_{e}})$ has isolated points in $(C L (X), τ_{W_{d}})$ . For $d = e$ , this implies a theorem of Costantini on topological completeness of $(C L (X), τ_{W_{d}})$ . We show that for nonseparable $X$ the hyperspace $(C L (X), τ_{W_{d}})$ may contain a closed copy of the rationals. This answers a question of Zsilinszky.

On the notion of potential for mappings between linear spaces. A generalized version of the Poincaré lemma

Tullio Valent (2003)

Bollettino dell'Unione Matematica Italiana

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An approach to the theory of linear differential forms in a radial subset of an (arbitrary) real linear space $X$ without a Banach structure is proposed. Only intrinsic (partially linear) topologies on $X$ are (implicitly) involved in the definitions and statements. Then a mapping $F : U \subseteq X \to Y$ , with $X$ , $Y$ real linear spaces and $U$ a radial subset of $X$ , is considered. After showing a representation theorem of those bilinear forms $〈 \cdot, \cdot 〉$ on $X \times Y$ for which $〈x, y〉 = 0$ $\forall x \in X$ $\Rightarrow y = 0$ , we observe that the...

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

Benjamín Espinoza, Verónica Martínez-de-la-Vega, Jorge M. Martínez-Montejano (2009)

Colloquium Mathematicae

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A hit-and-miss topology ( $τ_{H M}$ ) is defined for the hyperspaces $2^{X}$ , Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between $τ_{H M}$ and the Vietoris topology and we find conditions on X for which these topologies are equivalent.

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...

The order topology for a von Neumann algebra

Emmanuel Chetcuti, Jan Hamhalter, Hans Weber (2015)

Studia Mathematica

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The order topology $τ_{o} (P)$ (resp. the sequential order topology $τ_{o s} (P)$ ) on a poset P is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra M we consider the following three posets: the self-adjoint part $M_{s a}$ , the self-adjoint part of the unit ball $M ¹_{s a}$ , and the projection lattice P(M). We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology...

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.

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.

Locally $G_{δ}$ -spaces

Zdeněk Frolík (1962)

Czechoslovak Mathematical Journal

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The quasi Isbell topology on function spaces

D. N. Georgiou, A. C. Megaritis (2015)

Colloquium Mathematicae

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In this paper, on the family (Y) of all open subsets of a space Y we define the so called quasi Scott topology, denoted by $τ_{q S c}$ . This topology defines in a standard way, on the set C(Y,Z) of all continuous maps of the space Y to a space Z, a topology $t_{q I s}$ called the quasi Isbell topology. The latter topology is always larger than or equal to the Isbell topology, and smaller than or equal to the strong Isbell topology. Results and problems concerning the topology $t_{q I s}$ are given.

Partial Hölder continuity results for solutions of non linear non variational elliptic systems with limit controlled growth

Luisa Fattorusso, Giovanna Idone (2002)

Bollettino dell'Unione Matematica Italiana

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Let $Ω$ be a bounded open subset of $R^{n}$ , $n > 4$ , of class $C^{2}$ . Let $u \in H^{2} (Ω)$ a solution of elliptic non linear non variational system $a (x, u, D u, H (u)) = b (x, u, D u)$ where $a (x, u, μ, ξ)$ and $b (x, u, μ)$ are vectors in $R^{N}$ , $N \geq 1$ , measurable in $x$ , continuous in $(u, μ, ξ)$ and $(u, μ)$ respectively. Here, we demonstrate that if $b (x, u, μ)$ has limit controlled growth, if $a (x, u, μ, ξ)$ is of class $C^{1}$ in $ξ$ and satisfies the Campanato condition $(A)$ and, together with $\frac{\partial a}{\partial ξ}$ , certain continuity assumptions, then the vector $D u$ is partially Hölder continuous for every exponent $α < 1 - \frac{n}{p}$ .