The Uniform Effros Property and Local Homogeneity

Kathryn F. Porter wrote a nice paper about several definitions of local homogeneity [Local homogeneity, JP Journal of Geometry and Topology 9 (2009), 129-136]. In this paper, she mentions that G. S. Ungar defined a uniformly locally homogeneous space [Local homogeneity, Duke Math. J. 34 (1967), 693-700]. We realized that this notion is very similar to what we call the uniform property of Effros [On Jones' set function T and the property of Kelley for Hausdorff continua, Topology Appl. 226 (2017), 51-65]. Here, we compare the uniform property of Effros with the uniform local homogeneity. We also consider other definitions of local homogeneity given in Porter's paper and compare them with the uniform property of Effros. We show that in the presence of compactness, the uniform property of Effros is equivalent to uniform local homogeneity and the local homogeneity according to Ho. With this result, we can change the hypothesis of the uniform property of Effros in Jones' and Prajs' decomposition theorems to uniform local homogeneity and local homogeneity according to Ho. We add to these two results the fact that the corresponding quotient space also has the uniform property of Effros.