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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:
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> > On Tuesday, December 11, 2012 10:13:06 PM UTC-6, William Elliot wrote:
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> > > (h,Y is a (Hausdorff) compactification of X when h:X -> Y is an embedding,
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> > > Y is a compact (Hausdorff) space and h(X) is a dense subset of Y.
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> > >
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> > > Why the extra luggage of the embedding for the definition of
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> > > compactification? Why isn't the definition simply
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> > >
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> > > Y is a compactification of X when there's some
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> > > embedding h:X -> Y for which h(X) is a dense subset of Y?
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> > Compactification usually carry an (implicit) universal property: given
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> > any compact space Z and continuous function g:X-->Z, there exists a
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> > unique continuous G:Y-->Z such that g = Gh.
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> Is that true of all compactifications, that they have the universal
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> 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
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> > useful in many circumstances.
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> 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
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> > > has the advantage of being simpler and more intuitive. So why is
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> > > it that the first is used in preference to the second which I've
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> > > seen used only in regards to one point compactifications?
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> > Why this is the only one you've seen is probably an artifact of where
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> > you've looked. I've seen both definitions.
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> 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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