Generalized homogeneity of finite and of countable topological spaces (Q1112362)

A topological space X is said to be homogeneous with respect to a class M of functions of X onto itself provided that for every two points \(p,q\in X\), there is an element \(f\in M\) with \(f(p)=q\). The authors consider three classes of functions, namely: homeomorphisms, open continuous surjections, and continuous surjections. They prove that for finite spaces, there is no difference between the three derived concepts of homogeneity. They also prove that a countable metrizable space is homogeneous if and only if it is homogeneous with respect to the class of all open continuous surjections.