First, Turing computability is generalized from finite to infinite sequences of symbols. Naming systems are called representations. Since only very few representations are of interest in applications, a very fundamental principle for defining effective representations for T 0 -spaces with countable bases is introduced. The concepts are applied to real numbers, open sets, compact sets and continuous functions. The problem of determining zeroes is considered.
Computational complexity is discussed and computability in measure theory is sketched.
The paper concludes with some remarks on other models for Computable Analysis.