Hereditarily normal, locally compact Dowker spaces (Q2701856)

As is well known a Dowker space is one which is normal but whose product with \([0,1]\) is not normal. Dowker spaces are notoriously hard to come by; so far the `nicest' (separable, first-countable or cardinality \(\aleph_1\)) Dowker spaces required extra set-theoretic hypotheses in their constructions. In this paper the author uses \(\lozenge\) to construct a hereditarily normal and locally compact Dowker space on the set~\(\omega_1\times\omega\), much in the spirit of \textit{P. de Caux}'s example [Topol. Proc., Vol. 1, Conf. Auburn Univ. 1976, 67-77 (1976; Zbl 0397.54019)], which itself was adapted by \textit{I. Juhász} [Consistency results in topology, in `Handbook of mathematical logic', (1978; Zbl 0443.03001), pp. 503-522] and \textit{M. E. Rudin} [Handbook of set-theoretic topology, 761-780 (1984; Zbl 0554.54005)]. It can be arranged to have the one-point compactification be Fréchet-Uryson and~\(\alpha_1\). By contrast the author points out how one can construct a non-normal hereditarily countably paracompact locally compact space from a \(Q\)-set.