When is a bornology on a uniformizable space induced by a uniformity?

Let $$X$$ be a metrizable topological space, and let $$B$$ be a nontrivial bornology on $$X$$. Sze-Tsen Hu showed in 1949 that $$B$$ is the collection of bounded sets with respect to some metric for the topology on $$X$$ if and only if $$B$$ has a countable base and for any $$S\in B$$, there exists a $$T\in B$$ such that the closure of $$S$$ is a subset of the interior of $$T$$. (See this journal paper.)

I’m interested in the analogous result for uniformities. That is, $$X$$ be a uniformizable topological space, AKA a completely regular space, and let $$B$$ be a nontrivial bornology on $$X$$. My question is, under what circumstances is $$B$$ the collection of bounded sets with respect to some uniformity for the topology on $$X$$.

Note that a subset $$A$$ of a uniform space is said to be bounded if for each entourage $$V$$, $$A$$ is a subset of $$V^n[F]$$ for some natural number $$n$$ and some finite set $$F$$.

1 Answer

It looks like this problem was solved by Tom Vroegrijk in this 2009 journal paper, 60 years after the analogous problem for metric spaces was solved.

Let $$X$$ be a uniformizable topological space, AKA a completely regular space, and let $$B$$ be a nontrivial bornology on $$X$$. Let's call a sequence $$(U_n)$$ of open sets a bounding sequence if the closure of $$U_n$$ is a subset of $$U_{n+1}$$ for all $$n$$ and every element of $$B$$ is a subset of some $$U_n$$. And let's call a set $$S$$ saturated if $$S$$ if for every bounding sequence $$(U_n)$$, $$S$$ is a subset of some $$U_n$$. Then $$B$$ is the collection of bounded sets with respect to some uniformity for the topology on $$X$$ if and only if $$B$$ contains every saturated set.