Elements of General Representation Theory

Abstract

1°. Let G be a topological group and let X be a Hausdorff topological space. We let Homeo X denote the group of all homeomorphisms X → X. An action of G on X is, by definition, a homomorphism A: G → Homeo X. which is strongly continuous, meaning that the map g → A(g)x, G → X, is continuous for every fixed x ∈ X. The range of this map is called the orbit of the point x (under the given action A) and is denoted by o(x). The set of all orbits, endowed with the quotient topology, is called the orbit space, or the quotient space of the action A and is denoted by X/A. If it reduces to a point, A is said to be transitive. and X is called a homogeneous space of the group G. We say that two points x₁, x₂ ∈ X are equivalent (under A) if O(x₁) = O(x₂), in which case we write, as is customary, x₁ ∿ x₂.