We present an extension of the classical isomorphic classification of the Banach spaces C([0,α]) of all real continuous functions defined on the nondenumerable intervals of ordinals [0,α]. As an application, we establish the isomorphic classification of the Banach spaces $C\left(2^{}\times [0,\alpha ]\right)$ of all real continuous functions defined on the compact spaces $2^{}\times [0,\alpha ]$, the topological product of the Cantor cubes $2^{}$ with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. Consequently,...

We say that a cardinal function $\phi$ reflects an infinite cardinal $\kappa$, if given a topological space $X$ with $\phi \left(X\right)\ge \kappa$, there exists $Y\in {\left[X\right]}^{\le \kappa }$ with $\phi \left(Y\right)\ge \kappa$. We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences...

Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of $L_{\lambda ⁺\omega }$, with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with $⟨D^{ℳ},{\prec }^{ℳ}⟩$ a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where $⟨D^{{ℳ}^{\text{'}}},{\prec }^{{ℳ}^{\text{'}}}⟩$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{\lambda ⁺\omega }$ is exactly ${\beth }_{\delta \left(\lambda \right)}$. It was proved in [BK71] that if ℵ₀ < λ < κ$areregularcardinalnumbers,thenthereisaforcingextension,preservingcofinalities,suchthatintheextension$2λ = κ$and\delta \left(\lambda \right)<\lambda ⁺⁺.Weimprovethisresultbyprovingthefollowing:Suppose\aleph ₀<\lambda <\theta \le \kappa arecardinalnumberssuchthat$∙ ${\lambda }^{<\lambda }=\lambda$; ∙ cf(θ) ≥ λ⁺ and ${\mu }^{\lambda }<\theta$ whenever μ < θ; ∙ ${\kappa }^{\lambda }=\kappa$. Then there...

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put ${}_{\lambda }\left(\alpha \right)=s\in \left(\alpha \right):s\left(0\right)=\lambda =min\left[s\right(\beta ):\beta <\alpha ]$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $\lambda ₀i>\lambda ₁>...>{\lambda }_{n-1}$ and ordinals $\alpha ₀,...,{\alpha }_{n-1}$ such that $\alpha =\alpha ₀+\cdots +{\alpha }_{n-1}$ and $f=f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i{\in }_{{\lambda }_i}\left({\alpha }_i\right)$. Under GCH we prove that if α < ω₂ then (i) ${}_{\omega }\left(\alpha \right)=s{\in }^{\alpha }\omega ,\omega ₁:s\left(0\right)=\omega$; (ii) if λ > cf(λ) = ω, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega ₁-closedin\alpha$; (iii) if cf(λ) = ω₁, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega -closedandsuccessor-closedin\alpha$; (iv) if cf(λ) > ω₁, ${}_{\lambda }\left(\alpha \right){=}^{\alpha }\lambda$. This yields a complete characterization of the classes (α) for all...

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $𝒮$ of morphisms in a locally presentable category $𝒞$ of structures, the orthogonal class of objects...

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $\alpha \le 2^{}$, a topological group G such that $G^{\gamma }$ is countably compact for all cardinals γ < α, but $G^{\alpha }$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M{A}_{countable}$. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M{A}_{countable}$. However, the question has...

The elementary theory of ⟨α;×⟩, where α is an ordinal and × denotes ordinal multiplication, is decidable if and only if $\alpha <{\omega }^{\omega }$. Moreover if ${|}_r$ and ${|}_l$ respectively denote the right- and left-hand divisibility relation, we show that Th $⟨{\omega }^{{\omega }^{\xi }}{;|}_r⟩$ and Th $⟨{\omega }^{\xi }{;|}_l⟩$ are decidable for every ordinal ξ. Further related definability results are also presented.

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

In this paper, we define the direct sum ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the H-property if and only if each $X_i$ has the H-property, and ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the Schur property if and only if each $X_i$ has the Schur property. Moreover, we also show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ is rotund if and only if each $X_i$ is rotund.

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.

Let a space $X$ be Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff nonsingle point spaces $X_{\alpha }$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau$. Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^{\tau }$ or ${\omega }^{\tau }$ and a cardinal $\tau$ is infinite and not countably cofinal.

The cardinal invariants $𝔥,𝔟,𝔰$ of $𝒫\left(\omega \right)$ are known to satisfy that ${\omega }_1\le 𝔥\le min\{𝔟,𝔰\}$. We prove that all inequalities can be strict. We also introduce a new upper bound for $𝔥$ and show that it can be less than $𝔰$. The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of $P_{\kappa }\left(X\right)$, where κ ⊆ X. Suppose μ < κ, ${\mu }^{<\mu }=\mu$, and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in ${\left[H\left(\lambda \right)\right]}^{\mu ⁺}$ which are internally club but not internally approachable.

We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space $K_{2n}$ such that $C\left(K_{2n}\right)$ has no uncountable (semi)biorthogonal sequence ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of at most 2n-1 points of $K_{2n}$, but has biorthogonal systems ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves...

We investigate the question whether a system ${\left(E_i\right)}_{i\in I}$ of homogeneous linear equations over $\mathbb{Z}$ is non-trivially solvable in $\mathbb{Z}$ provided that each subsystem ${\left(E_j\right)}_{j\in J}$ with $\left|J\right|\le c$ is non-trivially solvable in $\mathbb{Z}$ where $c$ is a fixed cardinal number such that $c<\left|I\right|$. Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., $I$ being finite). (b) The answer is ‘No’ in the denumerable case (i.e., $\left|I\right|={\aleph }_0$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c\le {\aleph }_0$ is ‘No...

We prove that ${♦}^{*}$ implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality $2^{{\aleph }_1}$ which has points $G_{\delta }$. In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define $X_M$ to be X ∩ M with topology generated by $U\cap M:U\in \cap M$. Suppose $X_M$ is homeomorphic to the irrationals; must $X=X_M$? We have partial results. We also answer a question of Gruenhage by showing that if $X_M$ is homeomorphic to the “Long Cantor Set”, then $X=X_M$.

We characterize, in terms of X, the extensional dimension of the Stone-Čech corona βX∖X of a locally compact and Lindelöf space X. The non-Lindelöf case is also settled in terms of extending proper maps with values in $I^{\tau }\setminus L$, where L is a finite complex. Further, for a finite complex L, an uncountable cardinal τ and a $Z_{\tau }$-set X in the Tikhonov cube $I^{\tau }$ we find a necessary and sufficient condition, in terms of $I^{\tau }\setminus X$, for X to be in the class AE([L]). We also introduce a concept of a proper absolute...

The symbol ${\left(X_I\right)}_{\kappa }$ (with κ ≥ ω) denotes the space $X_I:={\prod }_{i\in I}{X}_i$ with the κ-box topology; this has as base all sets of the form $U={\prod }_{i\in I}{U}_i$ with $U_i$ open in $X_i$ and with $|i\in I:U_i\ne X_i|<\kappa$. The symbols w, d and S denote respectively the weight, density character and Suslin number. Generalizing familiar classical results, the authors show inter alia: Theorem 3.1.10(b). If κ ≤ α⁺, |I| = α and each $X_i$ contains the discrete space 0,1 and satisfies $w\left(X_i\right)\le \alpha$, then $w\left(X_{\kappa }\right)={\alpha }^{<\kappa }$. Theorem 4.3.2. If $\omega \le \kappa \le \left|I\right|\le 2^{\alpha }$ and $X={\left(D\left(\alpha \right)\right)}^I$ with D(α) discrete, |D(α)| = α, then $d\left({\left(X_I\right)}_{\kappa }\right)={\alpha }^{<\kappa }$. Corollaries 5.2.32(a)...

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies ${\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\partial }_j{f}_j{\left(\epsilon \right)\left|\right|}^{p}d\mu \left(\epsilon \right){\le }^p{\int }_{-1,1ⁿ}{\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\delta }_j\Delta f_j\left(\epsilon \right){\left|\right|}^{p}d\mu \left(\epsilon \right)d\mu \left(\delta \right)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${{\partial }_j}_{j=1}^n$ and $\Delta ={\sum }_{j=1}^n{\partial }_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by ${ⁿ}_p\left(X\right)$, we show that ${ⁿ}_p\left(X\right)\le {\sum }_{k=1}^{n}1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${\left({\alpha }_j\right)}_{j=1}^{\infty }$ of scalars, there exists a subsequence ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ such that either every subsequence of ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ defines a universal series, or no subsequence of ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ defines a universal series. In particular examples we decide which of the two cases holds.

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different...

We show that given infinite sets $X,Y$ and a function $f:X\to Y$ which is onto and $n$-to-one for some $n\in \mathbb{N}$, the preimage of any ultrafilter $ℱ$ of $Y$ under $f$ extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model $ℳ$ with a set of atoms $A$ and a finite-to-one onto function $f:A\to \omega$ such that for each free ultrafilter of $\omega$ its preimage under $f$ does not extend to an ultrafilter. In addition, we show that in $ℳ$ there exists an ultrafilter compact...

Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection $a_{\alpha }:\alpha \in \subset A$ there exist α ≠ β ∈ such that $a_{\alpha }\in a_{\beta }{A}^{}$ is isomorphic to ${⨁}_{i=1}^r(ℂ\left[X\right]/X^{d_i}ℂ\left[X\right])\oplus E$, where $d₁,...,d_r\in \mathbb{N}$, and E is either $Xℂ\left[X\right]/X^{d₀}ℂ\left[X\right]$ for some d₀ ∈ ℕ or a 1-dimensional ${⨁}_{i=1}^rℂ\left[X\right]/X^{d_i}ℂ\left[X\right]$-bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence ${\varphi }_{ℳ}$ if ${\varphi }_{ℳ}$ has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if ${\aleph }_{\beta }$ is characterized by a Scott sentence, then $2^{{\aleph }_{\beta +\beta ₁}}$ is (homogeneously) characterized by a Scott sentence, for all 0 <...

Let $f:X\to Y$ be a continuous map such as an open map, a closed map or a quotient map. We study under what circumstances $f$ remains an open, closed or quotient map in forcing extensions.

We construct, under Axiom ♢, a family ${\left(C\left(K_{\xi }\right)\right)}_{\xi <2^{\left(2^{\omega }\right)}}$ of indecomposable Banach spaces with few operators such that every operator from $C\left(K_{\xi }\right)$ into $C\left(K_{\eta }\right)$ is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size $2^{\omega }$ instead of $2^{\left(2^{\omega }\right)}$.

Given an uncountable cardinal κ with $\kappa ={\kappa }^{<\kappa }$ and $2^{\kappa }$ regular, we show that there is a forcing that preserves cofinalities less than or equal to $2^{\kappa }$ and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein...