(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?