fom
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Re: Stone Cech
Posted:
Mar 17, 2013 8:11 PM


On 3/17/2013 8:35 AM, David C. Ullrich wrote: > On Sun, 17 Mar 2013 01:28:41 0700, William Elliot <marsh@panix.com> > wrote: > >> On Fri, 15 Mar 2013, David C. Ullrich wrote: >>> On Thu, 14 Mar 2013 19:49:08 0700, William Elliot <marsh@panix.com> >>> wrote: >>> >>>> Let (g,Y) be a Cech Stone compactification of S. >>>> If (f,X) is a compactification of S, does X embed in Y? >>> >>> Isn't this clear from the universal property of the SC >>> compactification? >> >> No. All I get is that there's a closed continuous surjection h >>from Y onto X for which hg(S) is injective. > > I didn't read the question carefully, sorry. Sort of assumed > it was what would seem like a sensible question regarding > the universal property of the SC compactification. > > The answer to the actual question is no, X does not embed in Y, > or at least "surely not  the defining property of the SC > compactification simply has nothing to do with spaces > embedding in Y". > > What's true is that X is a _quotient space_ of Y. > >> >>>> If (g,Y) is a compactification of S and >>>> for all compactifications (f,X), X embeds in Y >>>> is (g,Y) a Stone Cech compactification of S? > > Similarly here  I misread the question as something about > whether another compactification sharing the same > universal property as the SC compactification must > be homeomorphic to the SC compactification (the > answer to _that_ is yes, and the proof starts as > I suggested). > > There's simply no reason to think that the answer to > the question you ask is yes, unless possibly it's yes > vacuously  I can't imagine an example of (g,Y) that > has the property you assume here. > > In particular, when you ask "If (g,Y) has property P, > must (g,Y) be a SC compactification of S?" I sort of > assumed that property P must be a property that > the SC compactification actually _satisfies_. > That's simply not so here  it's not true that every > compactification of S embeds in the SC compactification. >
So, how does your last statement reconcile with the text I quoted from Munkres that describes StoneCech compactification as maximal "in some sense".
I realize there is a lot in the context of definitions for terms. So I am curious how you mean this.

