On 12/11/2012 10:13 PM, 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? > > 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? >
What if one can describe another mapping g:X->Y for which the image g(X) is not dense in Y?
Perhaps the language for one-point compactification reflects the fact that the topology of the resulting compact space arises from the existing open and compact sets from the given locally compact Hausdorff space.
It is intrinsically well-defined.
I am sure others will give you more useful opinions.