(i) unions of effective sequences of disjoint algebraic open intervals;

(ii) unions of effective sequences of algebraic open intervals;

(iii) unions of effective sequences of rational open intervals;

For the equivalence (i), the While language must be augmented by a strong (Kleene) OR operator, and for equivalences (ii) and (iii) it must be further augmented by a strong existential quantifier over the naturals (ExistN).

We also show that the class of While+(ExistN) semicomputable relations on reals is closed under projections.

The proof makes essential use of the continuity of the operations of the algebra.