Characterizations of semicomputable sets of real numbers

We give some characterizations of semicomputability of sets of reals by programs in certain While programming languages over a topological partial algebra of reals. We show that such sets are semicomputable if and only if they are one of the following:(i)unions of effective sequences of disjoint algebraic open intervals;(ii)unions of effective sequences of rational open intervals;(iii)unions of effective sequences of algebraic open intervals. For the equivalence (i), the While language must be augmented by a strong OR operator, and for equivalences (ii) and (iii) it must be further augmented by a strong existential quantifier over the naturals (While∃N).We also show that the class of While∃N semicomputable relations on reals is closed under projection. The proof makes essential use of the continuity of the operations of the algebra.