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Topic: Precompactness
Replies: 9   Last Post: Dec 13, 2012 4:19 PM

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Precompactness
Posted: Dec 13, 2012 2:52 AM
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On Wed, 12 Dec 2012, Kaba wrote:
> 12.12.2012 5:24, William Elliot wrote:
> > On Tue, 11 Dec 2012, Kaba wrote:

> > > Let X be a locally compact Hausdorff space. Is every open set of X
> > > precompact (compact closure)?

> Related, let X be a Hausdorff space. Royden (Real analysis) defines E subset X
> to be _bounded_ if it is contained in a compact set. It seems to me that
> precompact and bounded are equivalent properties.

It's not for R with the include 0 topology, { U | 0 in U } \/ {empty set}.
{0} is bounded but not precompact.

> Assume E is precompact. Then cl(E) is a compact set which contains E.
> Therefore E is bounded. Assume E is bounded. Then there is a compact set K
> such that E subset K. Since X is Hausdorff, K is closed.

Why is X Hausdorff?

> subset K. Since cl(E) is a closed subset of a compact set K, cl(E) is compact.
> Therefore E is precompact.
> Unless I am missing something?

Assuming X is Hausdorff.

For Hausdorff spaces, or more general, kc spaces (compact sets are
closed), precompact and bounded are equivalent. Otherwise, they're not.

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