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?
> >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.
>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.