Compactification and completion as absolute closure

Abstract:

It is well known that a Tychonoff (respectively, Hausdorff uniform) space is compact (resp., complete) iff it is absolutely closed, i.e., dense in no other such space; we shall sketch a proof of this to our purpose, which is: Given $X$, we apply Zorn’s Lemma to obtain a space maximal with respect to the property of containing $X$ densely, thus compact or complete. For uniform spaces, the “maximal extension” is automatically the completion; for Tychonoff spaces, we must, and do, explicitly arrange things so that the “maximal extension” has the universal mapping property describing the Stone-Čech compactification. A variant of the construction yields the Hewitt realcompactification. A crucial point in the proofs is (of course) the exhibition of an upper bound for a chain; this is, in essence, a direct limit construction.