Topological Transformation Groups: Selected Topics

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{mapping}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.