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.