The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Precompactness
Posted: Dec 13, 2012 4:19 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Dec 13, 1:52 am, 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:
> > > > 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.

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.