Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.
|
|
Math Forum
»
Discussions
»
sci.math.*
»
sci.math
Notice: We are no longer accepting new posts, but the forums will continue to be readable.
Topic:
Cech Stone Compactification
Replies:
7
Last Post:
Jan 2, 2013 3:43 AM
|
 |
|
A N Niel
Posts:
2,255
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 > 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.
|
|
|
|