A note on bornologies

TL;DR

This paper explores the equivalence of various conditions characterizing bornologies on sets, linking them to vector topologies, uniformities, metrics, and a new concept called antitall bornologies, with topological and functional characterizations.

Contribution

It establishes the equivalence of multiple conditions for bornologies, introduces the concept of antitall bornologies, and provides their topological and functional characterizations.

Findings

Equivalence of conditions for bornologies involving vector topologies and uniformities

Introduction and characterization of antitall bornologies

Connections between bornologies and metric/discrete structures

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.