The Hurewicz property is a classical generalization of o-compactness. Sierpi & nacute;ski sets (whose existence follows from CH) are standard examples of non-o-compact Hurewicz spaces. We show, solving a problem stated by Szewczak and Tsaban [19], that for each Sierpi & nacute;ski set S of cardinality at least b there is a Hurewicz space H with SXH not Hurowicz. Some other questions in the literature concerning this topic are also answered. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, Al training, and similar technologies.

For any cardinal number κ and an index set Γ, the Σκ-product of real lines Σκ(ℝΓ) consists of all elements of ℝΓ having < κ many nonzero coordinates. A compact space K is κ-Corson compact if it can be embedded into Σκ(ℝΓ) for some Γ. The class of (ω1)-Corson compact spaces has been intensively studied over the last decades. We discuss properties of κ-Corson compacta for various cardinal numbers κ as well as properties of related Boolean algebras and spaces of continuous functions. We present here a detailed discussion of the class of ω-Corson compacta extending the results of Nakhmanson and Yakovlev [NY]. For κ > ω, our results on κ-Corson compact spaces are related to the line of research originated by Kalenda [Ka2] and Bell and Marciszewski [BM], and continued by Bonnet, Kubiś and Todorčević in their recent paper [BKT].

A topological space Y has the property (B) of Banakh if there is a countable family {A_n:n∈ℕ} of closed nowhere dense subsets of Y absorbing all compact subsets of Y. In this note we show that the space C_p(X) of continuous real–valued functions on a Tychonoff space X with the topology of pointwise convergence, fails to satisfy the property (B) if and only if the space C_p(X) satisfies the κ –Fréchet–Urysohn property. Additionally, we provide an analogous characterization for the compact–open topology on C(X). Finally, we give examples of Tychonoff spaces X whose all bounded subsets are finite, yet X fails to have the property (κ ) . This answers a question of Tkachuk.

Given a Tychonoff space $X$, we call a sequence $\langle\mu_n\colon n\in\omega\rangle$ of signed Borel measures on $X$ a finitely supported Josefson--Nissenzweig sequence (in short a JN-sequence) if: 1) for every $n\in\omega$ the measure $\mu_n$ is a finite combination of one-point measures and $\|\mu_n\|=1$, and 2) $\int_Xf\,\mathrm{d}\mu_n\to0$ for every continuous function $f\in C(X)$. Our main result asserts that if a Tychonoff space $X$ admits a JN-sequence, then there exists a JN-sequence $\langle\mu_n\colon n\in\omega\rangle$ such that: i) $\mbox{supp}(\mu_n)\cap\mbox{supp}(\mu_k)=\emptyset$ for every $n\neq k\in\omega$, and ii) the union $\bigcup_{n\in\omega}\mbox{supp}(\mu_n)$ is a discrete subset of $X$. We also prove that if a Tychonoff space $X$ carries a JN-sequence, then either there is a JN-sequence $\langle\mu_n\colon n\in\omega\rangle$ on $X$ such that $|\mbox{supp}(\mu_n)|=2$ for every $n\in\omega$, or for every JN-sequence $\langle\mu_n\colon n\in\omega\rangle$ on $X$ we have $\lim_{n\to\infty}|\mbox{supp}(\mu_n)|=\infty$.

For a free filter F on ω , endow the space N_F=ω∪{p_F} , where p_F∉ω , with the topology in which every element of ω is isolated whereas all open neighborhoods of p_F are of the form A∪{p_F} for A∈ F . Spaces of the form N_F constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space N_F carries a sequence ⟨μ _n:n∈ω⟩ of normalized finitely supported signed measures such that μ _n(f)→ 0 for every bounded continuous real-valued function f on N_F if and only if F^*≤ _K𝒵 , that is, the dual ideal F^* is Katětov below the asymptotic density ideal 𝒵 . Consequently, we get that if F^*≤ _K𝒵 , then: (1) if X is a Tychonoff space and N_F is homeomorphic to a subspace of X, then the space C_p^*(X) of bounded continuous real-valued functions on X contains a complemented copy of the space c_0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and N_F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck.

Cembranos and Freniche proved that for every two infinite compact Hausdorff spaces X and Y the Banach space C ( X × Y ) of continuous real-valued functions on X × Y endowed with the supremum norm contains a complemented copy of the Banach space c 0 . We extend this theorem to the class of C p -spaces, that is, we prove that for all infinite Tychonoff spaces X and Y the space C p ( X × Y ) of continuous functions on X × Y endowed with the pointwise topology contains either a complemented copy of ℝ ω or a complemented copy of the space ( c 0 ) p = {( x n ) n ∈ ω ∈ ℝ ω : x n → 0}, both endowed with the product topology. We show that the latter case holds always when X × Y is pseudocompact. On the other hand, assuming the Continuum Hypothesis (or even a weaker set-theoretic assumption), we provide an example of a pseudocompact space X such that C p ( X × X ) does not contain a complemented copy of ( c 0 ) p . As a corollary to the first result, we show that for all infinite Tychonoff spaces X and Y the space C p ( X × Y ) is linearly homeomorphic to the space C p ( X × Y ) × ℝ, although, as proved earlier by Marciszewski, there exists an infinite compact space X such that C p ( X ) cannot be mapped onto C p ( X ) × ℝ by a continuous linear surjection. This provides a positive answer to a problem of Arkhangel’ski for spaces of the form C p ( X × Y ). Another corollary—analogous to the classical Rosenthal-Lacey theorem for Banach spaces C ( X ) with X compact and Hausdorff—asserts that for every infinite Tychonoff spaces X and Y the space C k ( X × Y ) of continuous functions on X × Y endowed with the compact-open topology admits a quotient map onto a space isomorphic to one of the following three spaces: ℝ ω , ( c 0 ) p or c 0 .

We study the question for which Tychonoff spaces $X$ and locally convex spaces $E$ the space $C_p(X,E)$ of continuous $E$-valued functions on $X$ contains a complemented copy of the space $(c_0)_p=\{x\in\mathbb{R}^\omega\colon x(n)\to0\}$, both endowed with the pointwise topology. We provide a positive answer for a vast class of spaces, extending classical theorems of Cembranos, Freniche, and Doma\'nski and Drewnowski, proved for the case of Banach and Fr\'echet spaces $C_k(X,E)$. Also, for given infinite Tychonoff spaces $X$ and $Y$, we show that $C_p(X,C_p(Y))$ contains a complemented copy of $(c_0)_p$ if and only if any of the spaces $C_p(X)$ and $C_p(Y)$ contains such a subspace.

We discuss two problems concerning the class Eberlein compacta, i.e., weakly compact subspaces of Banach spaces. The first one deals with preservation of some classes of scattered Eberlein compacta under continuous images. The second one concerns the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. We show that the existence of such Eberlein compacta is consistent with ZFC. We also show that it is consistent with ZFC that each Eberlein compact space of weight $$> \omega _1$$ contains a nonmetrizable closed zero-dimensional subspace.

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space X X , the function space C p ( X ) C_p(X) is not countable dense homogeneous. This answers a question posed recently by R. Hernández-Gutiérrez. We also conclude that, for any infinite-dimensional Banach space E E (dual Banach space E ∗ E^\ast ), the space E E equipped with the weak topology ( E ∗ E^\ast with the weak ∗ ^\ast topology) is not countable dense homogeneous. We generalize some results of Hrušák, Zamora Avilés, and Hernández-Gutiérrez concerning countable dense homogeneous products.

Sciendo provides publishing services and solutions to academic and professional organizations and individual authors. We publish journals, books, conference proceedings and a variety of other publications.

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact $K_{\mathcal A}$ generated by an almost disjoint family $\mathcal A$ of infinite subsets of $\omega$ -- we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space $C(K_{\mathcal A})$ is uniquely determined up to isomorphism by the cardinality of $\mathcal A$ whenever $|{\mathcal A}|<{\mathfrak c}$, while there are $2^{\mathfrak c}$ nonisomorphic spaces $C(K_{\mathcal A})$ with $|{\mathcal A}|= {\mathfrak c}$. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of $c_0$ and some $C(K)$ spaces.

We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the continuum hypothesis $c_0$ has a nontrivial twisted sum with every space of the form $Y=C(K)$, where $K$ is compact and not metrizable. This gives a consistent positive solution to a problem posed by Cabello, Castillo, Kalton and Yost.

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact KA generated by an almost disjoint family A of infinite subsets of ω — we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin’s axiom the space C(KA) is uniquely determined up to isomorphism by the cardinality of A whenever |A| < c, while there are 2c nonisomorphic spaces C(KA) with |A| = c. We also investigate Koszmider’s problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of c0 and some C(K) spaces.

We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the continuum hypothesis $c_0$ has a nontrivial twisted sum with every space of the form $Y=C(K)$, where $K$ is compact and not metrizable. This gives a consistent positive solution to a problem posed by Cabello, Castillo, Kalton and Yost.

In this paper we develop a technique of constructing uni- formly continuous maps between function spaces Cp(X) endowed with the pointwise topology. We prove that if a space X is compact metrizable and strongly countable-dimensional, then there exists a uniformly contin- uous surjection from Cp([0,1]) onto Cp(X). We provide a partial result concerning the reverse implication. We also show that, for every infinite Polish zero-dimensional space X, the spaces Cp(X) and Cp(X) x Cp(X) are uniformly homeomorphic. This partially answers two questions posed by Krupski and Marciszewski.

We investigate the following problem posed by Cabello Sanchez, Castillo, Kalton, and Yost: Let K be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of co and C(K), i.e., does there exist a Banach space X containing a non -complemented copy Y of co such that the quotient space X/Y is isomorphic to C(K)? Using additional set -theoretic assumptions we give the first examples of compact spaces K providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either K is the Cantor cube 2(omega 1) or K is a separable scattered compact space of height 3 and weight omega(1), then every twisted sum of co and C(K) is trivial. We also construct nontrivial twisted sums of co and C(K) for K belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces K subset of L which do not admit an extension operator C(K) -> C(L). (C) 2017 Elsevier Inc. All rights reserved.

We give an example of an infinite metrizable space X such that the space C p (X), of continuous real-valued functions on X endowed with the pointwise topology, is not homeomorphic to its own square C p (X) × C p (X). The space X is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A. V. Arhangel’skii.

Abstract The first good message is to the effect that people possess reason as a source of intellectual insights, not available to the senses, as e.g. axioms of arithmetic. The awareness of this fact is called rationalism. Another good message is that reason can daringly quest for and gain new plausible insights. Those, if suitably checked and confirmed, can entail a revision of former results, also in mathematics, and - due to the greater efficiency of new ideas - accelerate science’s progress. The awareness that no insight is secured against revision, is called fallibilism. This modern fallibilistic rationalism (Peirce, Popper, Gödel, etc. oppose the fundamentalism of the classical version (Plato, Descartes etc.), i.e. the belief in the attainability of inviolable truths of reason which would forever constitute the foundations of knowledge. Fallibilistic rationalism is based on the idea that any problem-solving consists in processing information. Its results vary with respect to informativeness and its reverse - certainty. It is up to science to look for highly informative solutions, in spite of their uncertainty, and then to make them more certain through testing against suitable evidence. To account for such cognitive processes, one resorts to the conceptual apparatus of logic, informatics, and cognitive science.

We study extension operators between spaces $\sigma_n(2^X)$ of subsets of $X$ of cardinality at most $n$. As an application, we show that if $B_H$ is the unit ball of a nonseparable Hilbert space $H$, equipped with the weak topology, then, for any $0<\lambda<\mu$, there is no extension operator $T: C(\lambda B_H)\to C(\mu B_H)$.

We start the systematic study of Fréchet spaces which are ℵ-spaces in the weak topology. A topological space X is an ℵ0-space or an ℵ-space if X has a countable k-network or a σ-locally finite k-network, respectively. We are motivated by the following result of Corson (1966): If the space Cc(X) of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology is a Banach space, then Cc(X) endowed with the weak topology is an ℵ0-space if and only if X is countable. We extend Corson's result as follows: If the space E:=Cc(X) is a Fréchet lcs, then E endowed with its weak topology σ(E,E′) is an ℵ-space if and only if (E,σ(E,E′)) is an ℵ0-space if and only if X is countable. We obtain a necessary and some sufficient conditions on a Fréchet lcs to be an ℵ-space in the weak topology. We prove that a reflexive Fréchet lcs E in the weak topology σ(E,E′) is an ℵ-space if and only if (E,σ(E,E′)) is an ℵ0-space if and only if E is separable. We show however that the nonseparable Banach space ℓ1(R) with the weak topology is an ℵ-space.