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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: 1,223
Registered: 1/8/12
Re: Cech Stone Compactification
Posted: Jan 1, 2013 4:40 AM
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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?



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