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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: 11,117
Registered: 12/4/04
Re: Compactification
Posted: Dec 13, 2012 1:12 PM
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On Thursday, December 13, 2012 2:15:23 AM UTC-6, William Elliot wrote:
> On Tue, 11 Dec 2012, Arturo Magidin wrote:
>

> > On Tuesday, December 11, 2012 10:13:06 PM 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, Oh Great Complainer When Other People Don't Understand Your Babblings?


> > 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**.


> > > I see no advantage to the first definition. The second definition
>
> > > has the advantage of being simpler and more intuitive. So why is
>
> > > it that the first is used in preference to the second which I've
>
> > > seen used only in regards to one point compactifications?
>
>
>

> > Why this is the only one you've seen is probably an artifact of where
>
> > you've looked. I've seen both definitions.
>
>
>
> The two part definition seems more popular.


Which hardly contradicts that your assertions are based on your lack of familiarity with reality.

--
Arturo Magidin



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