Hausdorff first countable, countably compact space is \(\omega\)-bounded (Q2474541)

The authors state that they use ``the axiomatics of Set Theory,'' usually denoted by ZFC, and claim in Corollary~1 (p. 172) that if a topological space is Hausdorff, first countable and countably compact then it is \(\omega\)-bounded. Unfortunately, Corollary~1 cannot be proved in ZFC because there are known models of ZFC in which there exists a Hausdorff, first countable and countably compact space that is not \(\omega\)-bounded. For example, see the space of \textit{Adam Ostaszewski} [J. Lond. Math. Soc., II. Ser. 14, 505--516 (1976; Zbl 0348.54014)]. Moreover, in their abstract, the authors claim that Corollary~1 solves Problem~288 in ``Open Problems in Topology'' [van Mill and Reed eds., North-Holland, Amsterdam (1990; Zbl 0718.54001)]. Since their Corollary~1 cannot be proved in ZFC, the authors have not solved Problem~288. The authors make two more errors. One occurs in Remark~7 (p. 173) where they claim the space \(\gamma{\mathbf N}\) given by Nyikos in [loc. cit.], is not first countable. It is well known that \(\gamma{\mathbf N}\) is first countable. Another error occurs in the claim on page 174, lines 17--18, that \(\omega\) is not dense in the space \(X_{\omega_1}\) constructed by \textit{P. Nyikos} in his article in [Handbook of Set-theoretic Topology, 663--684 (1987; Zbl 0583.54002)]. It is known that \(\omega\) is dense in \(X_{\omega_1}\). The authors' Claim~2 (p. 174) states that there is a \(T_1\) (although not Hausdorff) first countable, countably compact space that is not \(\omega\)-bounded. It should be noted, however, that an example with these properties was given earlier by Nyikos with the following simple description: ``refine the cofinite topology on \(\omega_1\) by declaring all initial segments to be open,'' see p. 129, lines 7--8 in the article by Nyikos in the book ``Open Problems in Topology,'' (ibid).