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

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Compactification
Posted: Dec 12, 2012 11:59 AM
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On Tue, 11 Dec 2012 20:13:06 -0800, 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 inuition 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?


>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?





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