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

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William Elliot

Posts: 1,553
Registered: 1/8/12
Re: Compactification
Posted: Dec 13, 2012 3:15 AM
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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? I wouldn't think so but haven't found a counter example.
Is there an example for the two point compactification [0,1] of R,
that shows it doesn't have the universal property?

> E.g. the Stone-Czech compactification is of this kind.

> This also provides universality and uniqueness, which makes it more
> useful in many circumstances.


Aren't two universal compactifications of the same space, homeomorphic?
How is uniqueness used? It didn't seem necessary for homeomorphism
of universal compactifications (of the space).

> > 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.



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