On Effros continua and the uniform property of Effros

Abstract

We recall a theorem by E. G. Effros about the actions of a separable complete metric group acting transitively on a complete metric space. We consider the definition, by D. P. Bellamy and K. F. Porter of an Effros continuum in the class of Hausdorff homogeneous continua. We also recall the definition of the uniform property of Effros for Hausdorff continua. We prove that a homogeneous Hausdorff continuum is an Effros continuum if and only if it has the uniform property of Effros. We consider the weak property of Effros introduced by F. W. Simmons and show that Hausdorff continua with the weak property of Effros are homogeneous. We introduce the uniform weak property of Effros. We show that it is equivalent to the definition given by Simmons and that a Hausdorff continuum with the uniform property of Effros has uniform the weak property of Effros.

Keywords

• Continuum
• decomposable continuum
• Effros continuum
• Hausdorff continuum
• metric continuum
• group action