The current implementation of the Neo-Darwinian model of evolution typically assumes that the set of possible phenotypes is organized into a highly symmetric and regular space equipped at least with a notion of distance, for example, a Euclidean vector space. Recent computational work on the biophysical genotype-phenotype model defined by the folding of RNA sequences into secondary structures suggests a rather different picture. If phenotypes are organized according to genetic accessibility, the resulting space lacks a metric and is formalized by an unfamiliar structure, known as a pretopology. If recombination is taken into account, an even weaker structure, known as neighborhood space, must be used. Patterns of phenotypic evolution — such as punctuation, irreversibility, and modularity — result naturally from the properties of the genotype-phenotype map, which, given the genetic accessibility structure, defines accessibility in the phenotype space. The classical framework, however, addresses these patterns by exclusively invoking natural selection on suitably imposed fitness landscapes. We extend the explanatory level for phenotypic evolution from fitness considerations alone to include the topological structure of phenotype space as induced by the genotype-phenotype map. The topological framework allows us to consider e.g. the continuity of an evolutionary trajectory in an unambiguous way. Lewontin’s notion of “quasi-independence” of characters can be formalized as the assumption that a region of the phenotype space is represented by a product space of orthogonal factors. In this picture each character corresponds to a factor of a region of the phenotype space. We consider any region of the phenotype space that has a given factorization as a “type”, i.e., as a set of phenotypes that share the same set of phenotypic characters. Using the notion of local factorizations a theory of character identity can be developed that is based the correspondence of local factors in different regions of the phenotype space.