The hereditary paracompactness of \(X^ 2\) and the metrizability of X (Q1109364)

The author proves that for a Lindelöf p-space X, the following are equivalent: (1) X is metrizable, (2) \(X^ 2\) is hereditarily paracompact, (3) \(X^ 2\setminus \Delta\) is paracompact, where \(\Delta\) is the diagonal in \(X^ 2\), (4) Every open cover of \(X^ 2\setminus \Delta\) has a locally finite open refinement consisting of sets of the form \(U\times V\), where U and V are open in X (a so-called rectangular open covering), (5) \(X^ 2\setminus \Delta\) has a locally finite rectangular open covering. That (1) implies (4) follows from the author's theorem that if \(X^ 2\) is hereditarily Lindelöf, then (4) holds for \(X^ 2\setminus \Delta\). The author's results extend those of \textit{G. Gruenhage} in [Topology Appl. 17, 287-304 (1984; Zbl 0547.54016)] who proved the equivalence of (1), (2), and (3) for compact \(T_ 2\)-spaces. In [Topology Appl. 28, 11-15 (1988; Zbl 0636.54025)] \textit{G. Gruenhage} and \textit{J. Pelant} have shown that if X is a regular \(\Sigma\)-space such that \(X^ 2\setminus \Delta\) is paracompact, then X has a \(G_{\delta}\)-diagonal and give as a corollary that a p-space (hence Tikhonov) X such that \(X^ 2\setminus \Delta\) is paracompact is metrizable. Thus (1), (2), and (3) are equivalent for p-spaces without using the Lindelöf property.