Stone-cech Compactifications via Adjunctions

The Stone-Cech compactification of a space X is described by adjoining to X continuous images of the Stone-tech growths of a complementary pair of subspaces of X. The compactification of an example of Potoczny from [P] is described in detail. The Stone-Cech compactification of a completely regular space X is a compact Hausdorff space ßX in which X is dense and C*-embedded, i.e. every bounded real-valued mapping on X extends to ßX. Here we describe BX in terms of the Stone-Cech compactification of one or more subspaces by utilizing adjunctions and completely regular reflections. All spaces mentioned will be presumed to be completely regular. If A is a closed subspace of X and /maps A into Y, then the adjunction space X Of Y is the quotient space of the topological sum X © Y obtained by identifying each point of A with its image in Y. We modify this standard definition by allowing A to be an arbitrary subspace of X and by requiring / to be a C*-embedding of A into Y. The completely regular reflection of an arbitrary space y is a completely regular space pY which is a continuous image of Y and is such that any realvalued mapping on Y factors uniquely through p Y. The underlying set of p Y is obtained by identifying two points of Y if they are not separated by some real-valued mapping on Y. The resulting set has the property that for each real-valued mapping /on Y, a unique real-valued function p(f) can be defined on pY that factors/through pY. The topology on p Y is taken to be the weakest topology so that all of the functions pif) so obtained are continuous. Lemma I. If A is a subspace of X and fis a C*-embedding of A into Y, then X is C*-embedded in piX L)f Y). Proof. The mappings required in the proof are illustrated in the diagram. The mappings p, and p2 are the compositions of the quotient map onA"® f with the embeddings of X and Y into X © Y and k is any real-valued mapping on X. We show that both p2 and p|p2[A'] are embeddings. Since / is an embedding, p2 is one-to-one. To show that p2 is open onto its range, it is Received by the editors March 13, 1975. AMS (MOS) subject classifications (1970). Primary 54B17, 54D35; Secondary 54D60.