Bounded Sets in Topological Spaces

Cristina Bors,María Ferrer,Salvador Hernández

doi:10.3390/axioms11020071

Abstract

Let G be a monoid that acts on a topological space X by homeomorphisms such that there is a point x0∈X with GU=X for each neighbourhood U of x0. A subset A of X is said to be G-bounded if for each neighbourhood U of x0 there is a finite subset F of G with A⊆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.

Highlights

Introduction and Basic FactsThe notion of a bounded subset is ubiquitous in many parts of mathematics, in functional analysis and topological groups
This provides a very general notion of boundedness that includes both the bounded subsets considered in functional analysis and in topological groups
It is proved 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, extending results obtained by Grothendieck for metrizable separable locally convex spaces [1], generalized subsequently by Burke and Todorcevicand, separately, Saxon and Sánchez-Ruiz for metrizable locally convex spaces [2,3] and by Chis, Ferrer, Hernández and Tsaban for metrizable groups [4,5]

Summary

Introduction and Basic Facts

The notion of a bounded subset is ubiquitous in many parts of mathematics, in functional analysis and topological groups. We approach this concept from a broader viewpoint. We consider the action of a monoid G on a topological space X and associate it with a canonical family of G-bounded subsets This provides a very general notion of boundedness that includes both the bounded subsets considered in functional analysis and in topological groups. We obtain sufficient conditions for a G-space X to be locally G-bounded, which applies to topological groups This provides the frame for extending to this setting some results by Burke and Todorcevicand, separately, Saxon and Sáchez-Ruiz Vilenkin [8] applied this general approach in the realm of topological groups

G-Spaces

G-Boundedness

Infinite Cardinals

Dense Subspaces

G-Barrelled Groups

Discussion