  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.
