TY - CHAP

T1 - Topological Transformation Groups

T2 - Selected Topics

AU - Megrelishvili, Michael

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=52049084890&partnerID=8YFLogxK

U2 - 10.1016/B978-044452208-5/50043-0

DO - 10.1016/B978-044452208-5/50043-0

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AN - SCOPUS:52049084890

SN - 9780444522085

SP - 423

EP - 437

BT - Open Problems in Topology II

PB - Elsevier

ER -