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

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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 [Korovina, Margarita V., and Oleg V. Kudinov, The Uniformity Principle for Σ–definability with Applications to Computable Analysis, In Proceedings of CiE'07, Lecture Notes in Computer Science 4497 (2007), 416–425; Korovina, Margarita V., Computational aspects of Σ-definability over the real numbers without the equality test, In Proceedings of CSL'03, Lecture Notes in Computer Science 2803 (2003), 330–344; Korovina, Margarita V., and Oleg V. Kudinov, Semantic characterisations of second-order computability over the real numbers, In Proceedings of CSL'01, Lecture Notes in Computer Science 2142 (2001), 160–172]. In [Korovina, Margarita V., and Oleg V. Kudinov, The Uniformity Principle for Σ–definability with Applications to Computable Analysis, In Proceedings of CiE'07, Lecture Notes in Computer Science 4497 (2007), 416–425] it has been shown that a set B⊂Rn is Σ–definable without the equality test if and only if B is c.e. open. If we allow the equality test, the structure of a Σ–definable subset of Rn 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 Rn? It this paper we give the negative answer to this question.