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A N Niel
Posts:
2,247
Registered:
12/7/04


Re: Cech Stone Compactification
Posted:
Dec 31, 2012 8:35 AM


In article <Pine.NEB.4.64.1212310124010.26424@panix2.panix.com>, 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. > * 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). >
So, in those more general spaces, the *construction* can still be carried out. But the map is not a homeomorphism. So the result of the construction is not a "compactification" in your sense.



