# Induced topology by a complete uniform space.

I know that Uniform space is generalization idea of metric space,Uniform space like metric space induce a topological space.

Now my question is ( or are ):- In case our Uniform space was complete ,what will be added to the induced topology, more specifically We know that Complete metric space induce paracompact topological space. Does Also this true for Uniform Space? that if the uniform space are complete are the topology space induced by the uniform space will be paracompact?

Remark:- A uniform space $X$ is complete if every Cauchy net/filter in $X$ converges (not necessarily to a unique point).

• All metric spaces, not just the complete ones, are paracompact. – Henno Brandsma Dec 7 '14 at 7:19
• ams.org/journals/proc/1977-062-02/S0002-9939-1977-0436085-3/… seems relevant. It gives a condition for uniform spaces to be paracompact. – Henno Brandsma Dec 7 '14 at 7:59
• – Henno Brandsma Dec 7 '14 at 8:02
• In the second link you put they are proving "Every paracompact Hausdorff space is completely uniformizable ". – Birkary Dec 7 '14 at 15:07
• It's only a counter example if it admits a complete uniformity – Henno Brandsma Dec 9 '14 at 6:25