On Sun, 17 Mar 2013 01:28:41 -0700, William Elliot <firstname.lastname@example.org> wrote:
>On Fri, 15 Mar 2013, David C. Ullrich wrote: >> On Thu, 14 Mar 2013 19:49:08 -0700, William Elliot <email@example.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 S-C >> compactification? > >No. All I get is that there's a closed continuous surjection h >from Y onto X for which h|g(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 S-C 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 S-C 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 S-C compactification must be homeomorphic to the S-C 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 S-C compactification of S?" I sort of assumed that property P must be a property that the S-C compactification actually _satisfies_. That's simply not so here - it's not true that every compactification of S embeds in the S-C compactification.
Possibly you've misunderstood the universal property of the S-C compactification. If not, you seem to be just making up random questions, asking whether a certain property characterizes the S-C 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 S-C >> 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 S-C 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.