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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>
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.
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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