Completely uniformizable space

[Moniker1998]

$X$ is completely uniformizable (or Dieudonne complete) if $X$ has an admissible uniform structure in which it's complete (as a uniform space)
link to definition on wikipedia

Theorems:
P162 realcompact => completely uniformizable [15.14a of G-J]
completely uniformizable + P164 non-measurable cardinality + P1 $T_0$ => P162 realcompact [15.20 of G-J]
P30 paracompact + P11 regular => completely uniformizable [listed on wikipedia]
completely uniformizable => P12 uniformizable

Comparison to theorems relating to P162 realcompact:
T386: P22 pseudocompact + P162 realcompact => P16 compact
stronger theorem: P22 pseudocompact + P1 $T_0$ + completely uniformizable => P16 compact [exercise 15.Q1 of G-J]
Question: Is above true without P1 $T_0$?

T382: P7 $T_4$ + P31 metacompact + P164 non-measurable cardinality => P162 realcompact
Question: P13 normal + P31 metacompact => completely uniformizable ?

For now it seems reasonable to leave those two theorems alone. Especially since an exercise is less credible than a reference in a book (even though exercises in G-J are quite reliable).

G-J: Rings of continuous functions by Gillman and Jerison.

[felixpernegger]

#1426