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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 3:41 AM
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On Wed, 12 Dec 2012, David C. Ullrich wrote:
> William Elliot <marsh@panix.com> 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?

>
> >I see no advantage to the first definition. The second definition
> >has the advantage of being simpler and more intuitive.

> I disagree.
>
> The intuition is that "Y is compact and X is a dense subset of Y".
> Except that X is not literally a subset of Y. Saying that the
> compactification is h explains clearly exactly how one is
> to think of X as a subset of Y even though it's not.
>
> Say X is the set of rationals in (0,1) and Y = [3,4].
> With your second definition, Y is a compactification
> of X. But there are many different things you could
> mean by that! What actual subset of Y is the
> one that you're identifying with X?


Why is that important? Different embeddings don't
change Y nor the denseness of the embedded X.

Of course, constructionists would require the embedding but
certainly others shouldn't need an explicated embedding.



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