Actions of Topological Groups on Topological Spaces

Abstract

In this short chapter we introduce an important topic of continuous actions of topological groups on topological spaces. No attempt is made at a systematic treatment of the subject; this would require a separate book. Some such books already exist (see, in particular, [530, 86]). Our goal is much more modest — to give the reader just the avour of the topic by establishing several important results on dyadicity or similar properties of compact spaces in this context. One of these results concerns compact Gδ-sets in quotient spaces of ω-balanced topological groups. Even the corollary dealing with the case of the quotient space itself is extremely interesting and highly non-trivial. Another theorem provides a deep insight into the topological structure of compact spaces admitting a continuous transitive action of an !-narrow topological group. In fact, all compact spaces just mentioned have the following strong property — they are Dugundji spaces. Our arguments require several topological facts which usually do not form a part of standard courses on general topology, so the first section of the chapter familiarizes the reader with the concepts of Dugundji spaces, 0- soft mappings, and nearly open mappings. We also develop further the techniques involving inverse spectra (in Chapter 4 we have already made the first steps in this direction). We also introduce some basic concepts and elementary results on actions of topological groups on topological spaces.