On the Topological Completion

Let A be a Tychonoff space. As is well known, the points of the Stone-Cech compactification ßX "are" the zero-set ultrafilters of X, and the points of the Hewitt real-compactification vX are the zero-set ultrafilters which are closed under countable intersection. It is shown here that a zeroset ultrafilter is a point of the Dieudonné topological completion SX iff the family of complementary cozero sets is a-discretely, or locally finitely, additive. From this follows a characterization of those dense embeddings X C Y such that each continuous metric space-valued function on X extends over Y, and a somewhat novel proof of the Katëtov-Shirota Theorem. All spaces shall be Tychonoff. It is most convenient to view the class of topologically complete spaces as the class 91(911) of closed subspaces of products from the class <3It of metrizable spaces, that is, as the epireflective hull of 91L (Dieudonné showed that a Tychonoff space X has a compatible complete uniformity iff X admits an embedding with closed range into a product of metrizable spaces [D].) The topological completion SX of a Tychonoff space X is the epireflection of X into 91(911), that is, 8X is the essentially unique topologically complete space containing X densely such that each continuous map/: X -* Z (Z G 91(911)), admits a continuous extension 8f: 8X -* Z. This universal mapping property is implied by the weaker one for maps into spaces in 911, by the standard technique used to show for ßX that the universal mapping property for maps to [0, 1] implies the property for maps to compact spaces. See, e.g., [W]. We shall use this fact below. SA' may be constructed as the closure of a suitable homeomorph of A in a large product of metrizable spaces, similar to the common construction of ßX (e.g., [W]). The following is a more useful construction for our purpose. It depends on knowledge of ßX. 1. Lemma. 8X = r){(ßf)~l(M)\f: X -» M continuous, M G 91t). (Here ßf: ßX —* ßM is the extension over the Stone-Cech compactifications.) We sketch a proof of 1. Let Y = DM{(ßf)~l(M)). Clearly, a continuous map/: X ~* M has the extension ßf\Y: Y —> M, so it suffices to show that Y G 91(911). Since 91(911) is productive and closed-hereditary, it is closed under intersection (seen by realizing an intersection as a diagonal in a product), so it Received by the editors March 3, 1975. AMS (MOS) subject classifications (1970). Primary 54B99, 54C45, 54D20, 54D35, 54E14.