On Thu, 13 Dec 2012, Arturo Magidin wrote: > On Thursday, December 13, 2012 2:15:23 AM 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? > > Which part of "usually" did you not understand,
How S^1, the 1-pt compactification of R and [0,1], the 2-pt compactifications of R can have the universal property.
> > > 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**. >