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

 Messages: [ Previous | Next ]
 Kaba Posts: 289 Registered: 5/23/11
Re: Precompactness
Posted: Dec 12, 2012 12:31 PM

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?

--
http://kaba.hilvi.org

Date Subject Author
12/11/12 Kaba
12/11/12 Virgil
12/11/12 quasi
12/11/12 Kaba
12/11/12 William Elliot
12/11/12 S4M
12/12/12 Kaba
12/13/12 William Elliot
12/13/12 Kaba
12/13/12 Butch Malahide