Remarks on Σ–definability without the equality test over the Reals

In this paper we study properties of Σ–definability over the reals without the equality test which is one of the main concepts in the logical approach to computability over continuous data. It has been shown that a set $B\subset R^n$ is Σ–definable without the equality test if and only if $ B $ is c.e. open. If we allow the equality test, the structur$e of a Σ–definable subset of $R^n$ can be rather complicated. The next natural question to consider is the following. Is there an effective procedure producing a set which is a maximal c.e. open subset of a given Σ–definable with the equality subset of $R^n$? It this paper we give the negative answer to this question