On Tue, 11 Dec 2012 20:13:06 -0800, William Elliot <firstname.lastname@example.org> 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.
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?