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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

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?

Date Subject Author
12/11/12 William Elliot
12/11/12 magidin@math.berkeley.edu
12/13/12 William Elliot
12/13/12 magidin@math.berkeley.edu
12/14/12 William Elliot
12/14/12 magidin@math.berkeley.edu
12/14/12 David Hartley
12/14/12 Butch Malahide
12/15/12 David Hartley
12/12/12 Shmuel (Seymour J.) Metz
12/13/12 William Elliot
12/13/12 Shmuel (Seymour J.) Metz
12/12/12 David C. Ullrich
12/13/12 William Elliot
3/17/13 fom
3/17/13 fom