The constructive completion of the space 𝒟(ℝ)

Abstract

AbstractWe prove in the framework of Bishop's constructive mathematics that the sequential completion \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \tilde {\cal D} $\end{document}(ℝ) of the space 𝒟(ℝ) is filter‐complete. Then it follows as a corollary that the filter‐completeness of 𝒟(ℝ) is equivalent to the principle BD‐ℕ, which can be proved in classical mathematics, Brouwer's intuitionistic mathematics and constructive recursive mathematics of Markov's school, but does not in Bishop's constructive mathematics. We also show that \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \tilde {\cal D} $\end{document}(ℝ) is identical with the filter‐completion which was provided by Bishop. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)