Projections and retractions

Abstract

The main result of this paper is a generalization of a theorem of R. S. Phillips [6] on the nonexistence of projections of l, onto c0. If S is a locally compact Hausdorff space then let C(S) denote the space of bounded continuous real (or complex) valued functions on S; also, let Co(S) be those functions in C(S) which vanish at infinity. If N is the space of positive integers with the discrete topology then l=0C(N) and co=Co(N). Thus, Phillips' theorem says that in the case where S= N there is no bounded projection of C(S) onto Co(S); that is, the space N does not have the projection property. It is natural to ask for a characterization of spaces with the projection property in terms of their topology. Unfortunately, we cannot achieve this but we do show in Theorem 2 that pseudocompactness is a necessary condition for this property (S is pseudocompact if and only if every real valued continuous function on S is bounded). However, as Example 3 demonstrates, pseudocompactness is not sufficient. As a corollary to Theorem 2 we obtain a result of W. W. Comfort [1 ]. Namely, if S is completely regular and there is a retraction of its Stone-Cech compactification O3S onto j3S\S then S is a locally compact peudocompact space. To establish his result Comfort appealed to a result of W. Rudin which depends on the continuum hypothesis. Hence, not only do we give a relatively simple proof of Comfort's theorem, but furthermore, we do so without using the continuum hypothesis. Before proceeding to the main theorem we will need the following theorem of I. Glicksberg [3]. The proof is not difficult and we shall not repeat it here.