Chapter 43 - Topological transformation groups: Selected topics

Abstract

This chapter discusses selected topics related to topological transformation groups. In the discussion presented, all topological spaces are Tychonoff. A topological transformation group, or a G-space is a triple (G,X, π ), wherein the continuous action of a topological group G on a topological space X is π : G ×X →X, π (g, x) := gx. Supposing that G acts on X1 and on X2, a continuous map f : X1 →X2 is a G-map (or, an equivariant map) if f(gx) =gf (x) for every (g, x) ∈G ×X1. The Banach algebra of all continuous real valued bounded functions, on a topological space X, is denoted by C(X). If (G,X, π) be a G-space, it induces the action G ×C(X) →C(X), with (gf)(x) =f(g−1x). A function f ∈C(X) is said to be right uniformly continuous, or also π-uniform, if the map G →C(X), g↦gf is norm continuous. Concepts related to equivariant compactifications and equivariant normality are also elaborated. Details of universal actions are also provided in the chapter.