Group-valued continuous functions with the topology of pointwise convergence

Shakhmatov, Dmitri; Spěvák, Jan

doi:10.1016/j.topol.2009.06.022

Cited by 23 publications

(49 citation statements)

References 15 publications

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“…Since every abelian topological group can be viewed as a topological Z-module, where Z is taken with the discrete topology, we immediately obtain the following result (for a definition of the free abelian topological group over a space X in a given class of topological groups see [12]). Corollary 6.3.…”

Section: Background On Free Topological Modulesmentioning

confidence: 95%

“…A topological property T is said to be preserved by G-equivalence within a given class C of topological spaces if for every pair X, Y of G-equivalent topological spaces such that X, Y ∈ C , the space X has property T whenever Y has it. Extending the well-established line of research in the classical C p -theory of Arhangel'skiȋ, in [12] the authors showed that many important topological properties and cardinal invariants are preserved by G-equivalence for certain classes of topological groups G (that include the real line R).…”

Section: Introductionmentioning

confidence: 91%

“…Following [12], we say that spaces X and Y are G-equivalent provided that the topological groups C p (X, G) and C p (Y, G) are topologically isomorphic.…”

Section: Introductionmentioning

confidence: 99%

“…Based on this observation, the following notion was introduced in [12]. A topological property T is said to be preserved by G-equivalence within a given class C of topological spaces if for every pair X, Y of G-equivalent topological spaces such that X, Y ∈ C , the space X has property T whenever Y has it.…”

Section: Introductionmentioning

confidence: 99%

“…We will need the notion of G ⋆⋆ -regularity from [12]. Given a topological group G, a topological space X is called G ⋆⋆ -regular provided that, whenever g ∈ G, x ∈ X and F is a closed subset of X satisfying x / ∈ F , there exists f ∈ C p (X, G) such that f (x) = g and f (F ) ⊆ {e}.…”

mentioning

confidence: 99%

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Module-valued functors preserving the covering dimension

Spěvák¹

2015

Commentationes Mathematicae Universitatis Carolinae

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Section: Background On Free Topological Modulesmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 91%

“…Following [12], we say that spaces X and Y are G-equivalent provided that the topological groups C p (X, G) and C p (Y, G) are topologically isomorphic.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Module-valued functors preserving the covering dimension

Spěvák¹

2015

Commentationes Mathematicae Universitatis Carolinae

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Direct sums and products in topological groups and vector spaces

Dikranjan

Shakhmatov

Spěvák

2016

Journal of Mathematical Analysis and Applications

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We call a subset $A$ of an abelian topological group $G$: (i) $absolutely$ $Cauchy$ $summable$ provided that for every open neighbourhood $U$ of $0$ one can find a finite set $F\subseteq A$ such that the subgroup generated by $A\setminus F$ is contained in $U$; (ii) $absolutely$ $summable$ if, for every family $\{z_a:a\in A\}$ of integer numbers, there exists $g\in G$ such that the net $\left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\}$ converges to $g$; (iii) $topologically$ $independent$ provided that $0\not \in A$ and for every neighbourhood $W$ of $0$ there exists a neighbourhood $V$ of $0$ such that, for every finite set $F\subseteq A$ and each set $\{z_a:a\in F\}$ of integers, $\sum_{a\in F}z_aa\in V$ implies that $z_aa\in W$ for all $a\in F$. We prove that: (1) an abelian topological group contains a direct product (direct sum) of $\kappa$-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality $\kappa$; (2) a topological vector space contains $\mathbb{R}^{(\mathbb{N})}$ as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains $\mathbb{R}^{\mathbb{N}}$ as its subspace if and only if it has an $\mathbb{R}^{(\mathbb{N})}$ multiplier convergent series of non-zero elements. We answer a question of Hu\v{s}ek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki

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On the strong Feller property for stochastic delay differential equations with singular drift

Bachmann

2020

Stochastic Processes and their Applications

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In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations, see [10]. The argumentation is based on the well-posedness and the strong Feller property of the equations' drift-free version. To this aim, we investigate a certain convergence of random variables in topological spaces in order to deal with discontinuous drift coefficients.Keywords: Stochastic delay differential equations, stochastic functional differential equation, strong Feller property, singular drift, Zvonkin's transformation. MSC2010: Primary 34K50; secondary 60B10, 60B12, 60H10. Notation 1.1. We denote by · OP and · HS the operator norm and respectively the Hilbert-Schmidt norm for matrices A ∈ R d×d , i.e.Additionally, we write for a, b ∈ [−∞, +∞] a ∧ b := min{a, b}, a ∨ b := max{a, b}.Notation 1.2. In the sequel, let r > 0 be an arbitrary but fixed number and define

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Group-valued continuous functions with the topology of pointwise convergence

Cited by 23 publications

References 15 publications

Module-valued functors preserving the covering dimension

Module-valued functors preserving the covering dimension

Direct sums and products in topological groups and vector spaces

On the strong Feller property for stochastic delay differential equations with singular drift

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