Is every completely regular topology induced by some proximity?

Question

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.

score 1 · Accepted Answer · answered Feb 12 '19 at 16:55

1

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.

answered Feb 12 '19 at 16:55

Henno Brandsma

250,824

Thanks. By the way, can you take a look at my question here? https://math.stackexchange.com/q/3110208/71829 – Keshav Srinivasan Feb 12 '19 at 17:23

Is every completely regular topology induced by some proximity?

1 Answers1