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Topic: Cech Stone Compactification
Replies: 7   Last Post: Jan 2, 2013 3:43 AM

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 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Cech Stone Compactification
Posted: Jan 1, 2013 4:40 AM

On Mon, 31 Dec 2012, Herman Rubin wrote:
> On 2012-12-31, William Elliot <marsh@panix.com> wrote:

> > Would somebody elucidate what Wikipedia was saying about
> > Stone-Cech compactification? It doesn't make sense for
> > isn't a compactification an embedding into an compact space.

>
> > Some authors add the assumption that the starting space be Tychonoff
> > (or even locally compact Hausdorff), for the following reasons:
> > * The map from X to its image in bX is a homeomorphism if and only
> > if X is Tychonoff.

> A space is Tychonov if and only if the inverse images of open
> sets under real-valued continuous functions form a base for
> the topology.

I don't see anything to prove. What defintion of bX is
being used that doesn't require X to be embedded in bX?

> > * The map from X to its image in bX is a homeomorphism to an open
> > subspace if and only if X is locally compact Hausdorff.

>
> If f is a bounded continuous function, the closure in bX of
> {x: f(x) <= c is compact for all c, and since X is an open
> subset of bX, there is, for each y in X, a bounded continuous
> function h which is 0 at y and 1 on bX\X. Then {h(y) < 1/2) gives
> the desired neighborhood whose closure in X is compact.

Ok, this I follow upon the presumption bX is Hausdorff.

> > The Stone-Cech construction can be performed for more general spaces
> > X, but the map X -> bX need not be a homeomorphism to the image of X
> > (and sometimes is not even injective).

What more general definitions of Cech Stone compactifications are
there. Any beyond the construction with filters and the construction
for Tychonov T0 spaces?

Date Subject Author
12/31/12 A N Niel
12/31/12 David C. Ullrich
12/31/12 William Elliot
12/31/12 Herman Rubin
1/1/13 William Elliot
1/1/13 David Hartley
1/2/13 William Elliot