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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm6301-9-2015,
author = {Mehrdad Karavan},
title = {Characterization of realcompactness and hereditary realcompactness in the class of normal nodec (submaximal) spaces},
journal = {Colloquium Mathematicae},
volume = {144},
year = {2016},
pages = {73-76},
zbl = {1341.54001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6301-9-2015}
}
Mehrdad Karavan. Characterization of realcompactness and hereditary realcompactness in the class of normal nodec (submaximal) spaces. Colloquium Mathematicae, Tome 144 (2016) pp. 73-76. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6301-9-2015/