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Abstract <p>The paper establishes the existence of a noncomplete constructive metric space complete with respect to each element of some class of measure 1 in the Cantor space, containing all Martin-Löf random sequences. It is proved that any constructive metric space defined in a standard way on an invariant set of constructive reals and complete with respect to each element of some class of measure 1 is complete. An example of a constructive metric space is constructed in which every fundamental sequence converges, but there exists no class of measure 1 such that the space is complete with respect to each of its elements.</p>

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On Complete and Almost Complete Constructive Metric Spaces

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • M. Kh. Faizrahmanov

摘要

Abstract

The paper establishes the existence of a noncomplete constructive metric space complete with respect to each element of some class of measure 1 in the Cantor space, containing all Martin-Löf random sequences. It is proved that any constructive metric space defined in a standard way on an invariant set of constructive reals and complete with respect to each element of some class of measure 1 is complete. An example of a constructive metric space is constructed in which every fundamental sequence converges, but there exists no class of measure 1 such that the space is complete with respect to each of its elements.