Date: Dec 31, 2012 8:35 AM
Author: A N Niel
Subject: Re: Cech Stone Compactification
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

> 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.

> * 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).

>

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.