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Re: Cech Stone Compactification
Posted:
Dec 31, 2012 2:16 PM
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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.
> * 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.
> 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).
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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