arXiv subjects

Arkady Leiderman

Publications and source records attributed to Arkady Leiderman.

17 recordsLinked to original sources

arXiv2025

Asplund spaces $C_k(X)$ beyond Banach spaces

This paper addresses the Asplund property for the space of continuous functions $C_k(X)$ equipped with the compact-open topology, when $X$ is an arbitrary Tychonoff space. Motivated by inconsistent definitions in prior literature extending the Asplund property beyond Banach spaces, we provide a unified and self-contained treatment of core results in this context. A characterization of the Asplund property for $C_k(X)$ is established, alongside a review of classical results, including the Namioka--Phelps theorem and its implications. All proofs are presented in a self-contained manner and rely on standard techniques.

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arXiv2024

On the product of Weak Asplund locally convex spaces

For locally convex spaces, we systematize several known equivalent definitions of Fr\'echet (G\^ ateaux) Differentiability Spaces and Asplund (Weak Asplund) Spaces. As an application, we extend the classical Mazur's theorem as follows: Let $E$ be a separable Baire locally convex space and let $Y$ be the product $\prod_{\alpha\in A} E_{\alpha}$ of any family of separable Fr\'echet spaces; then the product $E \times Y$ is Weak Asplund. Also, we prove that the product $Y$ of any family of Banach spaces $(E_{\alpha})$ is an Asplund locally convex space if and only if each $E_{\alpha}$ is Asplund. Analogues of both results are valid under the same assumptions, if $Y$ is the $\Sigma$-product of any family $(E_{\alpha})$.

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arXiv2024

Dense metrizable subspaces in powers of Corson compacta

We characterize when the countable power of a Corson compactum has a dense metrizable subspace and construct consistent examples of Corson compacta whose countable power does not have a dense metrizable subspace. We also give several remarks about ccc Corson compacta and, as a byproduct, we obtain a new proof of Kunen and van Mill's characterization of when a Corson compactum supporting a strictly positive measure is metrizable.

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arXiv2023

On $\Delta$-spaces

$\Delta$-spaces have been defined by a natural generalization of a classical notion of $\Delta$-sets of reals to Tychonoff topological spaces; moreover, the class $\Delta$ of all $\Delta$-spaces consists precisely of those $X$ for which the locally convex space $C_p(X)$ is distinguished. The aim of this article is to better understand the boundaries of the class $\Delta$, by presenting new examples and counter-examples. 1) We examine when trees considered as topological spaces equipped with the interval topology belong to $\Delta$. In particular, we prove that no Souslin tree is a $\Delta$-space. Other main results are connected with the study of 2) $\Psi$-spaces built on maximal almost disjoint families of countable sets; and 3) Ladder system spaces. It is consistent with CH that all ladder system spaces on $\omega_1$ are in $\Delta$. We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on $\omega_1$ which is not in $\Delta$. We resolve several open problems posed in the literature.

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arXiv2022

When is a locally convex space Eberlein-Grothendieck?

In this paper we undertake a systematic study of those locally convex spaces $E$ such that $(E, w)$ is (linearly) Eberlein-Grothendieck, where $w$ is the weak topology of $E$. Let $C_{k}(X)$ be the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology. The main results of our paper are: (1) For a first-countable space $X$ (in particular, for a metrizable $X$) the locally convex space $(C_{k}(X), w)$ is Eberlein-Grothendieck if and only if $X$ is both $σ$-compact and locally compact; (2) $(C_{k}(X), w)$ is linearly Eberlein-Grothendieck if and only if $X$ is compact. We characterize $E$ such that $(E, w)$ is linearly Eberlein-Grothendieck for several other important classes of locally convex spaces $E$. Also, we show that the class of $E$ for which $(E, w)$ is linearly Eberlein-Grothendieck preserves linear continuous quotients. Various illustrating examples are provided.

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arXiv2021

A note on Banach spaces $E$ admitting a continuous map from $C_p(X)$ onto $E_{w}$

$C_p(X)$ denotes the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the topology of pointwise convergence. A Banach space $E$ equipped with the weak topology is denoted by $E_{w}$. It is unknown whether $C_p(K)$ and $C(L)_{w}$ can be homeomorphic for infinite compact spaces $K$ and $L$ \cite{Krupski-1}, \cite{Krupski-2}. In this paper we deal with a more general question: what are the Banach spaces $E$ which admit certain continuous surjective mappings $T: C_p(X) \to E_{w}$ for an infinite Tychonoff space $X$? First, we prove that if $T$ is linear and sequentially continuous, then the Banach space $E$ must be finite-dimensional, thereby resolving an open problem posed in \cite{Kakol-Leiderman}. Second, we show that if there exists a homeomorphism $T: C_p(X) \to E_{w}$ for some infinite Tychonoff space $X$ and a Banach space $E$, then (a) $X$ is a countable union of compact sets $X_n, n \in ω$, where at least one component $X_n$ is non-scattered; (b) $E$ necessarily contains an isomorphic copy of the Banach space $\ell_{1}$.

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arXiv2021

Is the free locally convex space $L(X)$ nuclear?

Given a class $\mathcal P$ of Banach spaces, a locally convex space (LCS) $E$ is called {\em multi-$\mathcal P$} if $E$ can be isomorphically embedded into a product of spaces that belong to $\mathcal P$. We investigate the question whether the free locally convex space $L(X)$ is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive. If $X$ is a Tychonoff space containing an infinite compact subset then, as it follows from the results of \cite{Aus}, $L(X)$ is not nuclear. We prove that for such $X$ the free LCS $L(X)$ has the stronger property of not being multi-Hilbert. We deduce that if $X$ is a $k$-space, then the following properties are equivalent: (1) $L(X)$ is strongly nuclear; (2) $L(X)$ is nuclear; (3) $L(X)$ is multi-Hilbert; (4) $X$ is countable and discrete. On the other hand, we show that $L(X)$ is strongly nuclear for every projectively countable $P$-space (in particular, for every Lindelöf $P$-space) $X$. We observe that every Schwartz LCS is multi-reflexive. It is known that if $X$ is a $k_ω$-space, then $L(X)$ is a Schwartz LCS \cite{Chasco}, hence $L(X)$ is multi-reflexive. We show that for any first-countable paracompact (in particular, metrizable) space $X$ the converse is true, so $L(X)$ is multi-reflexive if and only if $X$ is a $k_ω$-space, equivalently, if $X$ is a locally compact and $σ$-compact space. Similarly, we show that for any first-countable paracompact space $X$ the free abelian topological group $A(X)$ is a Schwartz group if and only if $X$ is a locally compact space such that the set $X^{(1)}$ of all non-isolated points of $X$ is $σ$-compact.

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arXiv2021

On linear continuous operators between distinguished spaces $C_p(X)$

As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $Δ$-space. If there exists a linear continuous surjective mapping $T:C_p(X) \to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) \to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,α])$, where $α$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,α]) \to C_p(Y)$ is given. We also observe that for every countable ordinal $α$ all closed linear subspaces of $C_p([1,α])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $Δ$-spaces we prove that a linear continuous surjection $T:C_p(X) \to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X \subset \mathbb{R}^n$).

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arXiv2021

Basic properties of $X$ for which spaces $C_p(X)$ are distinguished

In our paper [18] we showed that a Tychonoff space $X$ is a $Δ$-space (in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $Δ$ of $Δ$-spaces is invariant under the basic topological operations. We prove that if $X \in Δ$ and $φ:X \to Y$ is a continuous surjection such that $φ(F)$ is an $F_σ$-set in $Y$ for every closed set $F \subset X$, then also $Y\in Δ$. As a consequence, if $X$ is a countable union of closed subspaces $X_i$ such that each $X_i\in Δ$, then also $X\in Δ$. In particular, $σ$-product of any family of scattered Eberlein compact spaces is a $Δ$-space and the product of a $Δ$-space with a countable space is a $Δ$-space. Our results give answers to several open problems posed in \cite{KL}. Let $T:C_p(X) \longrightarrow C_p(Y)$ be a continuous linear surjection. We observe that $T$ admits an extension to a linear continuous operator $\widehat{T}$ from $R^X$ onto $R^Y$ and deduce that $Y$ is a $Δ$-space whenever $X$ is. Similarly, assuming that $X$ and $Y$ are metrizable spaces, we show that $Y$ is a $Q$-set whenever $X$ is. Making use of obtained results, we provide a very short proof for the claim that every compact $Δ$-space has countable tightness. As a consequence, under Proper Forcing Axiom (PFA) every compact $Δ$-space is sequential. In the article we pose a dozen open questions.

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arXiv2020

A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications

We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $Δ$-space in the sense of \cite {Knight}. As an application of this characterization theorem we obtain the following results: 1) If $X$ is a Čech-complete (in particular, compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered. 2) For every separable compact space of the Isbell--Mrówka type $X$, the space $C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals $[0,ω_1]$, then $C_p(X)$ is not distinguished. We observe that the existence of an uncountable separable metrizable space $X$ such that $C_p(X)$ is distinguished, is independent of ZFC. We explore also the question to which extent the class of $Δ$-spaces is invariant under basic topological operations.

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arXiv2016

Countable Successor Ordinals as Generalized Ordered Topological Spaces

A topological space $L$ is called a linear ordered topological space (LOTS) whenever there is a linear order $\leq$ on $L$ such that the topology on $L$ is generated by the open sets of the form $(a, b)$ with $a < b$ and $a, b \in L \cup \{ -\infty, +\infty \}$. A topological space $X$ is called a generalized ordered space (GO-space) whenever $X$ is topologically embeddable in a LOTS. Main Theorem: Let $X$ be a Hausdorff topological space. Assume that any continuous image of $X$ is a GO-space. Then $X$ is homeomorphic to a countable successor ordinal (with the order topology). The converse trivially holds.

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arXiv2016

$ω^ω$-Dominated function spaces and $ω^ω$-bases in free objects of Topological Algebra

A topological space $X$ is defined to have an $ω^ω$-base if at each point $x\in X$ the space $X$ has a neighborhood base $(U_α[x])_{α\inω^ω}$ such that $U_β[x]\subset U_α[x]$ for all $α\leβ$ in $ω^ω$. We characterize topological and uniform spaces whose free (locally convex) topological vector spaces or free (Abelian or Boolean) topological groups have $ω^ω$-bases.

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arXiv2016

$\mathfrak G$-bases in free (locally convex) topological vector spaces

We characterize topological (and uniform) spaces whose free (locally convex) topological vector spaces have a local $\mathfrak G$-base. A topological space $X$ has a local $\mathfrak G$-base if every point $x$ of $X$ has a neighborhood base $(U_α)_{α\inω^ω}$ such that $U_β\subset U_α$ for all $α\leβ$ in $ω^ω$. To construct $\mathfrak G$-bases in free topological vector spaces, we exploit a new description of the topology of a free topological vector space over a topological (or more generally, uniform) space.

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arXiv2016

Linear continuous surjections of $C_{p}$-spaces over compacta

Let $X$ and $Y$ be compact Hausdorff spaces and suppose that there exists a linear continuous surjection $T:C_{p}(X) \to C_{p}(Y)$, where $C_{p}(X)$ denotes the space of all real-valued continuous functions on $X$ endowed with the pointwise convergence topology. We prove that $\dim X=0$ implies $\dim Y = 0$. This generalizes a previous theorem \cite[Theorem 3.4]{LLP} for compact metrizable spaces. Also we point out that the function space $C_{p}(P)$ over the pseudo-arc $P$ admits no densely defined linear continuous operator $C_{p}(P) \to C_{p}([0,1])$ with a dense image.

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arXiv2015

Density character of subgroups of topological groups

A subspace Y of a separable metrizable space X is separable, but without X metrizable this is not true even If Y is a closed linear subspace of a topological vector space X. K.H. Hofmann and S.A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, locally compact abelian groups and connected locally compact groups and is closed under products and closed subgroups. A topological group G is almost connected if the quotient group of G by the connected component of its identity is compact. We prove that an almost connected pro-Lie group is separable iff its weight is not greater than c. It is deduced that an almost connected pro-Lie group is separable if and only if it is a subspace of a separable Hausdorff space. It is proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. Every precompact topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact group of weight c. This implies that there is a wealth of closed nonseparable subgroups of separable pseudocompact groups. An example is presented under CH of a separable countably compact abelian group which contains a non-separable closed subgroup. It is proved that the following conditions are equivalent for an omega-narrow topological group G: (i) G is a subspace of a separable regular space; (ii) G is a subgroup of a separable topological group; (iii) G is a closed subgroup of a separable pathconnected locally pathconnected group.

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arXiv2014

The strong Pytkeev property in topological spaces

A topological space $X$ has the strong Pytkeev property at a point $x\in X$ if there exists a countable family $\mathcal N$ of subsets of $X$ such that for each neighborhood $O_x\subset X$ and subset $A\subset X$ accumulating at $x$, there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. We prove that for any $\aleph_0$-space $X$ and any space $Y$ with the strong Pytkeev property at a point $y\in Y$ the function space $C_k(X,Y)$ has the strong Pytkeev property at the constant function $X\to \{y\}\subset Y$. If the space $Y$ is rectifiable, then the function space $C_k(X,Y)$ is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces $(X_n,*_n)$, $n\inω$, with the strong Pytkeev property their Tychonoff product and their small box-product both have the strong Pytkeev property at the distinguished point. We prove that a sequential rectifiable space $X$ has the strong Pytkeev property if and only if $X$ is metrizable or contains a clopen submetrizable $k_ω$-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.

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arXiv2010

Uniform Eberlein compactifications of metrizable spaces

We prove that each metrizable space (of cardinality less or equal to continuum) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class of compact spaces, that contain the empty set, the singleton, and is closed under producing the Aleksandrov compactification of the topological sum of a family of compacta from that class.

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