A note on bornologies

Abstract: A bornology on a set $X$ is a family $\mathcal{B}$ of subsets of $X$ closed under taking subsets, finite unions and such that $\cup \mathcal{B}=X$. We prove that, for a bornology $\mathcal{B}$ on $X$, the following statements are equivalent: (1) there exists a vector topology $\tau$ on the vector space $\mathbb{V} (X) $ over $\mathbb{R}$ such that $\mathcal{B}$ is the family of all subsets of $X$ bounded in $\tau$; (2) there exists a uniformity $\mathcal{U}$ on $X$ such that $\mathcal{B}$ is the family of all subsets of $X$ totally bounded in $\mathcal{U}$; (3) for every $Y \subseteq X$, $Y \notin \mathcal{B}$, there exists a metric $d$ on $X$ such that $\mathcal{B}\subseteq \mathcal{B}_d$, $Y\notin \mathcal{B}_d$, where $\mathcal{B}_d$ is the family of all closed discrete subsets of $(X, d)$; (4) for every $Y \subseteq X$, $Y \notin \mathcal{B}$, there exists $Z\subseteq Y$ such that $Z^{\prime} \notin \mathcal{B}$ for each infinite subset $Z^{\prime}$ of $Z$. A bornology $\mathcal{B}$ satisfying $(4)$ is called antitall. We give topological and functional characterizations of antitall bornologies.