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

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A N Niel

Posts: 2,245
Registered: 12/7/04
Re: Cech Stone Compactification
Posted: Dec 31, 2012 8:35 AM
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In article <Pine.NEB.4.64.1212310124010.26424@panix2.panix.com>,
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.
> * 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.
>
> 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).
>


So, in those more general spaces, the *construction* can still be
carried out. But the map is not a homeomorphism. So the result of the
construction is not a "compactification" in your sense.



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