On Wed, 12 Dec 2012, David C. Ullrich wrote: > 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. > I disagree. > > The intuition 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?
Why is that important? Different embeddings don't change Y nor the denseness of the embedded X.
Of course, constructionists would require the embedding but certainly others shouldn't need an explicated embedding.