Topological Transformation Groups: Selected Topics

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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.

Original languageEnglish
Title of host publicationOpen Problems in Topology II
Number of pages15
ISBN (Print)9780444522085
StatePublished - 2007


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