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Re: Cech Stone Compactification
Posted:
Dec 31, 2012 9:16 PM


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 > >StoneCech compactification? It doesn't make sense for > >isn't a compactification an embedding into an compact space. > > The StoneCech 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 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). > >



