Induced topology by a complete uniform space.

Asked Dec 07 '14 at 03:00

Active Dec 07 '14 at 03:17

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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).

edited Dec 07 '14 at 03:17

asked Dec 07 '14 at 03:00

Birkary

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All metric spaces, not just the complete ones, are paracompact. – Henno Brandsma Dec 07 '14 at 07:19
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http://www.ams.org/journals/proc/1977-062-02/S0002-9939-1977-0436085-3/S0002-9939-1977-0436085-3.pdf seems relevant. It gives a condition for uniform spaces to be paracompact. – Henno Brandsma Dec 07 '14 at 07:59
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Also: http://math.stackexchange.com/a/154119/4280 – Henno Brandsma Dec 07 '14 at 08:02
In the second link you put they are proving "Every paracompact Hausdorff space is completely uniformizable ". – Birkary Dec 07 '14 at 15:07
Yes, but you asked for the reverse, right? – Henno Brandsma Dec 07 '14 at 15:08
Sorry, I just want to say, they are proving the stronger version of it, I mean, in my case the space is not necessary Hausdorff. – Birkary Dec 07 '14 at 15:11
more precisely,Can the Paracompact space admit a compatible complete unifomty ? – Birkary Dec 07 '14 at 15:17
Normally, uniform spaces are assumed to be $T_0$ (and then they are Tychonoff), and paracompact spaces are assumed to be Hausdorff (and thus normal). Inspect the proofs if the assumptions are necessary, if you're interested in the more general case. – Henno Brandsma Dec 08 '14 at 19:55
And if we had a non-normal Tychonoff space that admitted a complete uniformity, we'd have a counterexample. Maybe the Tychonoff plank? – Henno Brandsma Dec 08 '14 at 19:56
You mean deleted Tychonff plank? – Birkary Dec 09 '14 at 00:14
It's also known by that name, yes. – Henno Brandsma Dec 09 '14 at 05:20
"the Encyclopedia of General Topology By K.P. Hart, Jun-iti Nagata, J.E" said Tychnoff plank is nether countable compact nor normal. So in this case its our counter example. can you please make your comments as an answer I am sure , people get this useful. – Birkary Dec 09 '14 at 06:06
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It's only a counter example if it admits a complete uniformity – Henno Brandsma Dec 09 '14 at 06:25

Induced topology by a complete uniform space.

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