We show that if n is a positive integer and $2^{\aleph ₀}\le \aleph ₙ$, then for every positive integer m and for every real constant c > 0 there are functions $f₁,...,f_{n+m}:ℝⁿ\to ℝ$ such that $(f₁,...,f_{n+m})\left(ℝⁿ\right)={ℝ}^{n+m}$ and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that $(f_{i₁},...,f_{iₙ})\left(y\right)=y+w$ for $y\in x+(-c,c)\times {ℝ}^{n-1}$.

Let $X$ be a continuum and $n$ a positive integer. Let $C_n\left(X\right)$ be the hyperspace of all nonempty closed subsets of $X$ with at most $n$ components, endowed with the Hausdorff metric. For $K$ compact subset of $X$, define the hyperspace ${C_n}_K\left(X\right)=\{A\in C_n\left(X\right):K\subset A\}$. In this paper, we consider the hyperspace $C_K^n\left(X\right)=C_n\left(X\right)/{C_n}_K\left(X\right)$, which can be a tool to study the space $C_n\left(X\right)$. We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility. ...

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).

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

Let us denote by $\Phi (\lambda ,\mu )$ the statement that $𝔹\left(\lambda \right)=D{\left(\lambda \right)}^{\omega }$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $ℂ$ in $𝔹\left(\lambda \right)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V}\subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and...

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 $𝔠$.

In set theory without the axiom of choice ($\mathrm{AC}$), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC${}^{\mathrm{LO}}$ (AC for linearly ordered families of nonempty sets)—and hence AC${}^{\mathrm{WO}}$ (AC for well-ordered families of nonempty sets)— $\mathrm{DC}(<\kappa )$ (where $\kappa$ is an uncountable regular cardinal), and “for every infinite set $X$, there is a bijection $f:X\to \{0,1\}\times X$”, implies the statement “there exists a field $F$ such that...

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...

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 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)...

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,...

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...

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$ describing properties of Dense(ℚ). These invariants satisfy ${}_{\mathbb{Q}}$ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ$.WecomparethemwiththeiranaloguesinthewellstudiedBooleanalgebra\left(\omega \right)/fin.Weshowthat$ℚ = p$,$ℚ = t$and$ℚ = i$,whereas$ℚ > h$and$ℚ > r$arebothshowntoberelativelyconsistentwithZFC.Wealsoinvestigatecombinatoricsoftheidealnwdofnowheredensesubsetsof,\mathbb{Q}.Inparticular,weshowthat$non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................