The topological viewpoint on spaces of phenotypes presented in Stadler et al. (J Theor Biol 213:241-274, 2001) is revisited, and a quantified version is proposed. While necessary probabilistic information can be encoded in a topological- like fashion, it turns out that it is not reflected adequately by the concept of continuity. We propose alternative models, but the behavior of maps make these models non-topological in fundamental ways.