On some kinds of completeness and semitopological groups with property (w∗)

We show that if X is a Tychonoff metacompact weakly pseudocompact countably sieve-complete space then X is & Ccaron;ech-complete. A metric space X is & Ccaron;ech-complete if and only if X is Telg & aacute;rsky complete. Then it follows that an Oxtoby complete metric space is not necessarily Telg & aacute;rsky complete. This answers Question 11.1 in [8]. We give an internal characterization of subgroups of products of first-countable semitopological groups which are paracompact sigma- spaces. We also discuss some properties of regular omega- balanced, locally omega- good semitopological groups with a qpoint, property (w*) w *) and Sm(G) ( G ) <=omega. omega. We finally give a sufficient condition under which a semitopological group is topologically isomorphic to a subgroup of a product of first-countable M 1-semitopological groups. (c) 2024 Elsevier B.V. All rights reserved.