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

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Cech Stone Compactification
Posted: Dec 31, 2012 11:45 AM
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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.

>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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