We construct examples of mappings and between locally connected continua such that and are near-homeomorphisms while is not, and is a near-homeomorphism, while and are not. Similar examples for refinable mappings are constructed.
We consider isometry groups of a fairly general class of non standard products of metric spaces. We present sufficient conditions under which the isometry group of a non standard product of metric spaces splits as a permutation group into direct or wreath product of isometry groups of some metric spaces.