Fundamental Groups of a Topological Transformation Group

Some properties of a path space and the fundamental group σ(X, x₀, G) of a topological transformation group (X, G, π) are described. It is shown that u(X, x₀, H) is a normal subgroup of u(X, x₀, G) if His a normal subgroup of G ; Let (X, G, π) be a transformation group with the open action property. If every identification map p : Σ( X, x, G) → σ(X, z, G) is open for each x ∈ X, then λ induces a homeo-morphism between the fundamental groups σ(X, x₀, G) and σ(X, y₀, G) where λ is a path from x₀ to y₀ in X ; The space σ(X, x₀, G) is an H-space if the identification map p: Σ(X, x₀, G) → σ(X, x₀, G) is open in a topological transformation group (X, G, π).