On uniquely homogeneous spaces, I

Abstract

It is shown that all uniquely homogeneous spaces are connected. We characterize the uniquely homogeneous spaces that are semitopological or quasitopological groups. We identify two properties of homogeneous spaces called skew-2-flexibility and 2-flexibility that are useful in studying unique homogeneity. We also construct a large family of uniquely homogeneous spaces with only trivial continuous maps.