A Dowker space from a Lusin set (Q1329770)

A normal space is called a Dowker space if its product with the closed unit interval [0,1] is not normal. The author presents a construction of a first countable, locally countable Dowker space of size \(\aleph_ 1\) from a Luzin set. Let \({\mathcal A}\) be a Luzin subset of the real line. The topology \(\rho\) on \(\omega \times {\mathcal A}\) is defined such that (i) \((\omega \times {\mathcal A}, \rho)\) is normal, and (ii) \((\omega \times {\mathcal A},\rho)\) is not countably metacompact. The necessity of the assumption that \({\mathcal A}\) is a Luzin set is also denoted.