Homogeneity properties with isometries and Lipschitz functions

We consider metric variants of homogeneity, countable dense homogeneity (CDH) and strong local homogeneity (SLH) by requiring that the homeomorphisms that witness the homogeneity be isometries, respectively bi-Lipschitz maps that are almost isometries: iso-homogeneity, iso-CDH, iso-SLH, L-homogeneity, LCDH, and LSLH. We prove metric versions of Bennett's theorem that SLH implies CDH for complete spaces, and we show that every separable Banach space is LCDH. As applications, we investigate how a number of standard examples of CDH spaces fare with respect to metric homogeneity. ©2010 Rocky Mountain Mathematics Consortium.