The \(\beta\)-space property in monotonically normal spaces and go-spaces (Q2499782)

The notion of a \(\beta\)-space was introduced by \textit{R. H. Hodel} [Duke Math. J. 39, 253--263 (1972; Zbl 0242.54027)] as a generalization of a semi-stratifiable space, and recently it was proved by \textit{G. Ying} and \textit{C. Good} [Commentat. Math. Univ. Carol. 42, 771--778 (2001; Zbl 1090.54504)] that a space is a \(\beta\)-space if and only if it is monotonically countably metacompact. In this paper, the authors examine the role of the \(\beta\)-space property in generalized ordered spaces (\(=\) GO-spaces) and, more generally, in monotonically normal spaces. Their main theorems are: (1) A GO-space is metrizable if and only if it is a \(\beta\)-space with a \(G_\delta\)-diagonal. It is to be noted that any LOTS with a \(G_\delta\)-diagonal is metrizable, while a GO-space with a \(G_\delta\)-diagonal may fail to be metrizable. (2) A GO-space is metrizable if and only if it is a quasi-developable \(\beta\)-space. (3) A monotonically normal space that is hereditarily a \(\beta\)-space is hereditarily paracompact. (4) A GO-space is metrizable if and only if it is perfect and each of its subspaces is a \(\beta\)-space. There is an appendix on non-Archimedean spaces in which the authors prove various results announced without proof by P. Nyikos.