A note on bornologies

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 $Y\subseteq X$, $Y\mathrm{\notin }\mathcal{B}$, 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 $Y\subseteq X$, $Y\mathrm{\notin }\mathcal{B}$, there exists $Z\subseteq Y$ such that $Z^{\mathrm{\prime }}\mathrm{\notin }\mathcal{B}$ for each infinite subset $Z^{\mathrm{\prime }}$ of $Z$. A bornology $\mathcal{B}$ satisfying $(4)$ is called antitall. We give topological and functional characterizations of antitall bornologies.