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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: 11,142
Registered: 12/4/04
Re: Compactification
Posted: Dec 11, 2012 11:33 PM
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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?


Compactifications 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.

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

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

> 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.

--
Arturo Magidin



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