
Re: Stone Cech
Posted:
Mar 17, 2013 9:35 AM


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.
Possibly you've misunderstood the universal property of the SC compactification. If not, you seem to be just making up random questions, asking whether a certain property characterizes the SC compactification when that property simply has nothing to do with it.
>> My guess is that there is a counterexample to exactly what you wrote >> here. The universal property of bS says that any map from S to a compact >> space lifts _uniquely_ to a map from bS to the same space. If you add >> the same uniqueness bit to your hypotheses on (g,Y) then surely the >> answer is yes; consider the case where (f,X) is the SC >> compactifiication to get started. >> >I didn't find any elevator nor even an esclator. >Are their stairs anywhere? > >What's is apparent that if h,k in C(Y,X) and >f = hg, f = kg, then h = k. > >> >Why in the heck is a compactification an embedding >> >function and a compact space? Wouldn't be simpler >> >to define a compactification of a space S, as a >> >compact space into which S densely embeds? >> >> No. For the same reasons as the last time you asked. Or consider this: >> Part of the definition of the SC compactification is the _uniqueness_ >> of the lift of a map from S to any compact space. Let's say the notion >> you're considering is a WEification. A WEification could correspond to >> two different compactifications in the usual sense; now the uniqueness >> of the lift can't even be defined, much less asserted. >> >For latter consideration. > >Perhaps you could illustrate with the five different >one to four point point compactifications of two open >end line segements.

