On Tue, 11 Dec 2012, Arturo Magidin wrote: > 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?
> Compactification 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.
Is that true of all compactifications, that they have the universal property? I wouldn't think so but haven't found a counter example. Is there an example for the two point compactification [0,1] of R, that shows it doesn't have the universal property?
> E.g. the Stone-Czech compactification is of this kind.
> This also provides universality and uniqueness, which makes it more > useful in many circumstances.
Aren't two universal compactifications of the same space, homeomorphic? How is uniqueness used? It didn't seem necessary for homeomorphism of universal compactifications (of the space).
> > 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.