The quasi Isbell topology on function spaces

D. N. Georgiou; A. C. Megaritis

Displaying similar documents to “The quasi Isbell topology on function spaces”

Fine and quasi connectedness in nonlinear potential theory

David R. Adams, John L. Lewis (1985)

Annales de l'institut Fourier

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If $B_{α, p}$ denotes the Bessel capacity of subsets of Euclidean $n$ -space, $α > 0$ , $1 < p < \infty$ , naturally associated with the space of Bessel potentials of $L^{p}$ -functions, then our principal result is the estimate: for $1 < α p \leq n$ , there is a constant $C = C (α, p, n)$ such that for any set $E$ $min {B_{α, p} (E \cap Q), B_{α, p} (E^{c} \cap Q)} \leq C \cdot B_{α, p} (Q \cap \partial_{f} E)$ for all open cubes $Q$ in $n$ -space. Here $\partial_{f} E$ is the boundary of the $E$ in the $(α, p)$ -fine topology i.e. the smallest topology on $c$ -space that makes the associated $(α, p)$ -linear potentials continuous there. As a consequence,...

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...

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.

Further new generalized topologies via mixed constructions due to Császár

Erdal Ekici (2015)

Mathematica Bohemica

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The theory of generalized topologies was introduced by Á. Császár (2002). In the literature, some authors have introduced and studied generalized topologies and some generalized topologies via generalized topological spaces due to Á. Császár. Also, the notions of mixed constructions based on two generalized topologies were introduced and investigated by Á. Császár (2009). The main aim of this paper is to introduce and study further new generalized topologies called $μ_{12}^{C}$ via mixed constructions...

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 $C o n v_{F} (ℝ) \approx ℝ \times$ .

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...

Locally $G_{δ}$ -spaces

Zdeněk Frolík (1962)

Czechoslovak Mathematical Journal

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Some properties of the topology of $α$ -sets

D. Andrijević (1984)

Matematički Vesnik

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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.

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.

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...

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

G. Lassner (1975)

Publications du Département de mathématiques (Lyon)

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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...

$ω$ H-sets and cardinal invariants

Alessandro Fedeli (1998)

Commentationes Mathematicae Universitatis Carolinae

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A subset $A$ of a Hausdorff space $X$ is called an $ω$ H-set in $X$ if for every open family $𝒰$ in $X$ such that $A \subset ⋃ 𝒰$ there exists a countable subfamily $𝒱$ of $𝒰$ such that $A \subset ⋃ {\bar{V} : V \in 𝒱}$ . In this paper we introduce a new cardinal function $t_{s θ}$ and show that $| A | \leq 2^{t_{s θ} (X) ψ_{c} (X)}$ for every $ω$ H-set $A$ of a Hausdorff space $X$ .