Bounded Sets in Topological Spaces

Let G be a monoid that acts on a topological space X by homeomorphisms such that there is a point x(0)& ISIN;X with GU=X for each neighbourhood U of x(0). A subset A of X is said to be G-bounded if for each neighbourhood U of x(0) there is a finite subset F of G with A & SUBE;FU. We prove that for a metrizable and separable G-space X, the bounded subsets of X are completely determined by the bounded subsets of any dense subspace. We also obtain sufficient conditions for a G-space X to be locally G-bounded, which apply to topological groups. Thereby, we extend some previous results accomplished for locally convex spaces and topological groups.