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

 Messages: [ Previous | Next ]
 Butch Malahide Posts: 894 Registered: 6/29/05
Re: Precompactness
Posted: Dec 13, 2012 4:19 PM

On Dec 13, 1:52 am, William Elliot <ma...@panix.com> wrote:
> 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.

More generally, for any space X in which every compact set has a
compact closure (this includes compact spaces and kc spaces), a subset
of X is precompact if and only if is bounded.

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