The Effros metric

The concepts of the Effros metric and the Effros property are extended in such a way that a semigroup M of surjective self mappings of a bounded metric space (in place of autohomeomorphism group) is used in the definitions. Relations between the Effros property for M and M-homogeneity are investigated. Special attention is paid to locally connected continua and the class C of all continuous mappings between them. It is shown that local absolute retracts, as well as locally connected curves, have the Effros property for C, while 2-dimensional locally connected continua do not have this property. A number of questions are asked.