A proximity space is a set endowed with a relation defining a notion of when two subsets are near or far apart. A proximity space induces a topology, and such a topology is always completely regular. But I’m wondering if the converse is true.
My question is, is every completely regular topology induced by some proximity? Or is being “proximatizable” a stronger condition than being completely regular?
My motive for asking this question, by the way, is to understand the relation between proximity spaces and uniform spaces. Because a topology is completely regular if and only if it is induced by a uniformity.
Yes, any completely regular space is induced by a proximity space and also by a uniformity. It's all the same class topologically. Engelking devotes one chapter in his book General Topology on these notions and equivalences.