Constructing balleans (Q2278230)

A ballean (or a coarse space) \((X,\mathcal{E})\) is a set \(X\) with a coarse structure \(\mathcal{E}\), where a coarse structure is a family of subsets of \(X \times X\) which contains the diagonal and is closed under taking subsets, inverses, products and finite unions (see [\textit{J. Roe}, Lectures on coarse geometry. Providence, RI: American Mathematical Society (2003; Zbl 1042.53027)] and [\textit{I. Protasov} and \textit{M. Zarichnyi}, General asymptology. L'viv: VNTL Publishers (2007; Zbl 1172.54002)]). In this paper, the authors introduce three constructions of balleans, which are called bornological products, bouquets and combs, from a family of balleans. Here, a bornological product is a generalization of the Cantor macrocube introduced by \textit{T. Banakh} and \textit{I. Zarichnyi} [Groups Geom. Dyn. 5, No. 4, 691--728 (2011; Zbl 1246.54023)]. The authors prove theorems on metrizability and normality of these three constructions. A bornology on a set \(X\) is a family of subsets of \(X\) containing all finite subsets of \(X\) and closed under taking finite unions and subsets. For \(E \subset X\times X\) and \(x \in X\), let \(E[x]=\{ y \in X : (x,y)\in E \}\). For a bornology \(\mathcal{B}\) on a set \(X\), a coarse structure \(\mathcal{E}\) on \(X\) is said to be compatible with \(\mathcal{B}\) if \(\{ E[x]: E \in \mathcal{E}, \,x \in X \}=\mathcal{B}\). The authors also study the smallest and the largest coarse structures compatible with a given bornology.