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Re: Stone Cech
Posted:
Mar 17, 2013 4:28 AM
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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 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.
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