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

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Kaba

Posts: 289
Registered: 5/23/11
Re: Precompactness
Posted: Dec 13, 2012 3:11 PM
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13.12.2012 9:52, William Elliot wrote:
> On Wed, 12 Dec 2012, Kaba wrote:
>> 12.12.2012 5:24, William Elliot wrote:
>>> On Tue, 11 Dec 2012, Kaba wrote:
>
>> 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.


Therefore R is not Hausdorff:) Interesting example though of what can
wrong without the Hausdorff requirement.

>> 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?


By definition.

>> 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.


That's a nice term to know, thanks.

--
http://kaba.hilvi.org



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