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Re: Cech Stone Compactification
Posted:
Jan 1, 2013 4:40 AM


On Mon, 31 Dec 2012, Herman Rubin wrote: > 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. I don't see anything to prove. What defintion of bX is being used that doesn't require X to be embedded in bX?
> > * 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. Ok, this I follow upon the presumption bX is 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).
What more general definitions of Cech Stone compactifications are there. Any beyond the construction with filters and the construction for Tychonov T0 spaces?



