Bounded Sets in Topological Spaces

Let G be a monoid that acts on topological space X by homeomorphisms such there is point x0∈X with GU=X for each neighbourhood U of x0. A subset said to G-bounded if x0 finite F A⊆FU. We prove metrizable and separable G-space X, the bounded subsets are completely determined any dense subspace. also obtain sufficient conditions locally G-bounded, which apply groups. Thereby, we extend some previous results accomplished convex spaces