Mappings of βX to X*

Abstract

This chapter is devoted to the investigation of various types of mappings of βX to X*. The chapter will be in four parts. We will first consider a sequence of preliminary results giving sufficient conditions to ensure that the continuous image of a dense, C*-embedded subs pace will be C*-embedded in the image of the whole space. These results will then be applied to study mappings of βX onto X* and in particular to obtain necessary conditions for the existence of a retraction of βX onto X*. Next, the growths of the compactifications of a locally compact space X will be characterized as the continuous images of X*. Finally we will investigate mappings of extremally disconnected spaces and apply our results to mappings of Stone-Ćech compactifications of discrete spaces. As a corollary we will obtain our third proof that ℕ* is not homogeneous.