Topological homogeneity (Q2874855)

This is a survey article on topological homogeneity and some of its variants (e.g., countable dense homogeneity; unique homogeneity). The authors impose very few assumptions on the spaces under consideration and are also interested in spaces with some sort of pathology. A significant part of the paper discusses various cardinal invariants of homogeneous spaces, associated to the weight, tightness, \(\pi\)-bases, pseudobases and related concepts. Most of results in this direction say that a homogeneous space with certain properties has suitably small (certain) cardinal invariant.NEWLINENEWLINEA part of the paper is devoted to the question of when a product space is homogeneous or whether a homogeneous space of some class admits a factor of some other (or the same) class. The authors also discuss the following issues and notions: (a) a connection between homogeneity and (semi)topological groups and their actions; (b) sufficient conditions under which a homogeneous space is a coset of a topological group; (c) power-homogeneity and Corson compacta; (d) countable dense and unique homogeneity. The paper contains a lot of theorems (mostly without proofs) and open problems.NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].