On Σ\Sigma-definability without equality over the real numbers

In Delzell (1982) it has been shown that for first-order definability over the reals there exists an effective procedure which by a finite formula with equality defining an open set produces a finite formula without equality that defines the same set. In this paper we prove that there exists no such procedure for $\Sigma$-definability over the reals. We also show that there exists even no uniform effective transformation of the definitions of -definable sets (i. e., $\Sigma$-formulas) into new definitions of $\Sigma$-definable sets in such a way that the results will define open sets, and if a definition defines an open set, then the result of this transformation will define the same set. These results highlight the important differences between $\Sigma$-definability with equality and $\Sigma$-definability without equality