For every proximity, does there exist a uniformity which generates this proximity?

Question

For every proximity, does there exist a uniformity which generates this proximity?

This question may be generalized for different generalizations of proximities and uniformities. In fact I need it for funcoids and reloids: for every funcoid $f$ does there exist a reloid $g$ which generates it (that is $f = (\operatorname{FCD})g$)?

Reloids are just filters on the set of binary relations (on some set).

Funcoids are essentially a generalization of proximities with only the following axioms:

• ¬(∅ δ X) and ¬(X δ ∅);
• (A∪B)δC ⇔ AδC ∨ BδC;
• Cδ(A∪B) ⇔ CδA ∨ CδB.

Read http://www.mathematics21.org/binaries/funcoids-reloids.pdf at http://www.mathematics21.org/algebraic-general-topology.html about my theory of funcoids and reloids.

Answer 1

See

http://www.vabo.cz/stranky/hoskova/clanky/clanek18.pdf

especially page 3, for necessary and sufficient conditions on uniformizability of a proximity space. Whether every proximity space is uniformizable thus depends upon exactly what set of axioms one takes for a proximity space. From my cursory glance, the "Strong Axiom 5" of the above reference is not included in e.g. the definition of proximity spaces given here:

http://en.wikipedia.org/wiki/Proximity_space

(Sorry; I am not familiar with funcoids and reloids.)