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Characterization of realcompactness and hereditary realcompactness in the class of normal nodec (submaximal) spaces

Volume 144 / 2016

Mehrdad Karavan Colloquium Mathematicum 144 (2016), 73-76 MSC: Primary 54A25, 03E55; Secondary 54B05. DOI: 10.4064/cm6301-9-2015 Published online: 22 February 2016


Is it true in ZFC that every normal submaximal space of non-measurable cardinality is hereditarily realcompact? This question (posed by O. T. Alas et al. (2002)) is given a complete affirmative answer, for a wider class of spaces. In fact, this answer is a part of a bi-conditional statement: A normal nodec space $X$ is hereditarily realcompact if and only if it is realcompact if and only if every closed discrete (or nowhere dense) subset of $X$ has non-measurable cardinality.


  • Mehrdad KaravanDepartment of Mathematics
    Persian Gulf University
    Mahini St.
    Bushehr, Iran
    Bushehr House of Mathematics
    Farhang St.
    Bushehr, Iran

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