On Fri, 15 Mar 2013, David C. Ullrich wrote: > On Thu, 14 Mar 2013 19:49:08 -0700, William Elliot <firstname.lastname@example.org> > 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.
> >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? > > 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.