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Re: Compactification
Posted:
Dec 14, 2012 2:46 AM
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On Thu, 13 Dec 2012, Arturo Magidin wrote:
> On Thursday, December 13, 2012 2:15:23 AM UTC-6, William Elliot wrote:
> >
> > > > (h,Y is a (Hausdorff) compactification of X when h:X -> Y is an embedding,
> > > > Y is a compact (Hausdorff) space and h(X) is a dense subset of Y.
> >
> > > > Why the extra luggage of the embedding for the definition of
> > > > compactification? Why isn't the definition simply
> > > > Y is a compactification of X when there's some
> > > > embedding h:X -> Y for which h(X) is a dense subset of Y?
> >
> > > Compactification usually carry an (implicit) universal property:
> > > given any compact space Z and continuous function g:X-->Z, there
> > > exists a unique continuous G:Y-->Z such that g = Gh.
> >
> > Is that true of all compactifications, that they have the universal
> > property?
>
> Which part of "usually" did you not understand,
How S^1, the 1-pt compactification of R
and [0,1], the 2-pt compactifications of R
can have the universal property.
> > > This also provides universality and uniqueness, which makes it more
> > > useful in many circumstances.
> >
> > Aren't two universal compactifications of the same space, homeomorphic?
>
> Well, duh. That's what "universality and uniqueness" provides. Not only
> are they homeomorphic, they are homeomorphic via a **unique**
> homeomorphism **that respects the embeddings**.
>
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