Isomorphism theorem for BSS recursively enumerable sets on real closed fields

The main result of this paper shows roughly that any r.e. set $S$ in $R^N$, $N le infty$, where $R$ is a real closed field, is isomorphic to $R^{dim S}$ by a bijection $phi$ which is decidable over $S$. Moreover the map $S mapsto phi$ is computable. Some related matters are also considered like the computability of r.e. maps, characterisation of the real closed fields with a r.e. set of infinitesimals, and the dimension of r.e. sets.