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

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Posts: 289
Registered: 5/23/11
Re: Precompactness
Posted: Dec 12, 2012 12:31 PM
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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)?

> Yes. How would you prove it?


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.

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. Therefore
cl(E) 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?


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