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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,
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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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> 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?
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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