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Re: Cech Stone Compactification
Posted:
Dec 31, 2012 2:16 PM


On 20121231, William Elliot <marsh@panix.com> wrote: > Would somebody elucidate what Wikipedia was saying about > StoneCech 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 realvalued 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 StoneCech 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)4946054 FAX: (765)4940558



