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

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Posts: 11,383
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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