Represented spaces are the topological spaces on which computations can be performed. We investigate the descriptive complexity of sets in represented spaces. First, we prove that the standard representation of a countably-based space preserves the effective descriptive complexity of sets, and we prove that some results from descriptive set theory on Polish spaces extend to arbitrary countably-based spaces. Secondly, we study the larger class of coPolish spaces (in particular the space of polynomials), and we show that their representation does not always preserve the complexity of sets. We relate this mismatch with the sequential aspects of the space.