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Re: Cech Stone Compactification
Posted:
Dec 31, 2012 9:16 PM
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On Mon, 31 Dec 2012, David C. Ullrich wrote:
> On Mon, 31 Dec 2012 01:28:36 -0800, William Elliot <marsh@panix.com>
>
> >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.
>
> The Stone-Cech compactification is not a compactification,
> by the usual definitions.
It isn't?? When then is the usual definition of compactification
other than an embedding into a 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).
>
>
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