The gap between the dimensions of countably paracompact spaces (Q952631)

The following general problem (in ZFC) is under consideration: \textit{for what subclass \(\mathcal K\) of normal spaces are there no other relations in \(\mathcal K\) between the dimensions \(\dim, \) ind and Ind besides (1) \(\dim \leq \text{Ind}\), (2) ind \( \leq\) Ind and (3) \(\dim = 0\) implies Ind \( = 0\) ?} The author proves that the class of separable, first countable, \(\omega_1\)-compact, countably paracompact, normal spaces is such a \(\mathcal K\). He observes also, with the help of some known examples, that the class of paracompact spaces would be such a \(\mathcal K\) if one constructs a paracompact space \(X_n\) with ind \(X_n = 0\) and \(\dim X_n = \) Ind\(X_n = n\) for any given natural number \(n\).