Given a coarse space $(X,ℰ)$ with the bornology $ℬ$ of bounded subsets, we extend the coarse structure $ℰ$ from $X\times X$ to the natural coarse structure on $(ℬ\setminus \{\varnothing \left\}\right)\times (ℬ\setminus \{\varnothing \left\}\right)$ and say that a macro-uniform mapping $f:(ℬ\setminus \{\varnothing \left\}\right)\to X$ (or $f:{\left[X\right]}^2\to X$) is a selector (or 2-selector) of $(X,ℰ)$ if $f\left(A\right)\in A$ for each $A\in ℬ\setminus \left\{\varnothing \right\}$ ($A\in {\left[X\right]}^2$, respectively). We prove that a discrete coarse space $(X,ℰ)$ admits a selector if and only if $(X,ℰ)$ admits a 2-selector if and only if there exists a linear order “$\le$" on $X$ such that the family of intervals $\left\{\right[a,b]:a,b\in X,a\le b\}$ is a base for the bornology $ℬ$.

Let $M(X,𝒜)$ ($M^{*}(X,𝒜)$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X,𝒜)$, let $M_K(X,𝒜)$ be the family of all $f\in M(X,𝒜)$ such that $\mathrm{coz}\left(f\right)$ is compact, and let $M_{\infty }(X,𝒜)$ be all $f\in M(X,𝒜)$ that $\{x\in X:|f\left(x\right)|\ge \frac{1}{n}\}$ is compact for any $n\in \mathbb{N}$. We introduce realcompact subrings of $M(X,𝒜)$, we show that $M^{*}(X,𝒜)$ is a realcompact subring of $M(X,𝒜)$, and also $M(X,𝒜)$ is a realcompact if and only if $(X,𝒜)$ is a compact measurable space. For every nonzero real Riesz map $\varphi :M(X,𝒜)\to ℝ$, we prove that there is an element $x_0\in X$ such that $\varphi \left(f\right)=f\left(x_0\right)$ for every $f\in M(X,𝒜)$ if $(X,𝒜)$ is a compact measurable space....

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 ...............................................................................................................................................................

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\left\{b_n\right\}$ of positive elements of $B$, there is a sequence $\left\{{\lambda }_n\right\}$ of positive reals, and $b\in B$, with ${\lambda }_n{b}_n\le b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “$\sigma$” obtains for a Riesz space of continuous real-valued functions $C\left(X\right)$. A basic result is: For discrete $X$, $C\left(X\right)$ has $\sigma$ iff the cardinal $\left|X\right|<𝔟$, Rothberger’s bounding number. Consequences...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...