
Re: Compactification
Posted:
Dec 13, 2012 1:12 PM


On Thursday, December 13, 2012 2:15:23 AM UTC6, William Elliot wrote: > On Tue, 11 Dec 2012, Arturo Magidin wrote: > > > On Tuesday, December 11, 2012 10:13:06 PM UTC6, 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?
Which part of "usually" did you not understand, Oh Great Complainer When Other People Don't Understand Your Babblings?
> > This also provides universality and uniqueness, which makes it more > > > useful in many circumstances. > > > > Aren't two universal compactifications of the same space, homeomorphic?
Well, duh. That's what "universality and uniqueness" provides. Not only are they homeomorphic, they are homeomorphic via a **unique** homeomorphism **that respects the embeddings**.
> > > 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. > > > > The two part definition seems more popular.
Which hardly contradicts that your assertions are based on your lack of familiarity with reality.
 Arturo Magidin

