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Topic: Compactification
Replies: 15   Last Post: Mar 17, 2013 6:11 AM

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magidin@math.berkeley.edu

Posts: 11,118
Registered: 12/4/04
Re: Compactification
Posted: Dec 14, 2012 12:49 PM
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On Friday, December 14, 2012 1:46:30 AM UTC-6, William Elliot wrote:
> 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.


First, that's a completely different question altogether: above you asked whether it is true that **every** compactification has a universal property. Since I never made that claim, and had stated they need not always do so (I said they **usually** do, but "usually" does not mean "always"), your questions reflected a lack of attention to what was written.

Now you change your question, pretending that it was what you said all along? And you have the chutzpah to complain about other people all the time, as you do?

The one-point compactification is a quotient of the 2-point compactification. Any map that extends to the one-point compactification will necessarily extend to the 2 point compactification.

--
Arturo Magidin




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