Lexicographic products of GO-spaces (Q1676542)

\textit{M. J. Faber} [Metrizability in generalized ordered spaces. Mathematical Centre Tracts. 53. Amsterdam: Mathematisch Centrum (1974; Zbl 0282.54017)] has shown that the lexicographic order topology on a product of paracompact linearly ordered topological spaces (LOTS) is paracompact. This paper extends that result to products of generalized ordered (or GO-) spaces in the following sense: Each GO-space \((X,<,\tau)\) can be embedded densely in a standard way in a minimal LOTS, \(X^*\subseteq X\times \{-1,0,1\}\). If \(X_\alpha\) is a GO-space for each ordinal \(\alpha<\gamma\), then the lexicographic product order on \(\hat{ X}=\prod\{X_\alpha^*:\alpha<\gamma\}\) is defined as usual and the lexicographic product of the GO-spaces \(X_\alpha\) is the order and the topology on \(X=\prod\{X_\alpha:\alpha<\gamma\}\) inherited as a subspace of \(\hat{ X}\). The main theorem of this paper then states that a lexicographic ordered product of paracompact GO-spaces is paracompact. A (rather technical) characterization is also given of when a lexicographic product of GO-spaces is actually a LOTS. Some examples involving the Sorgenfrey line, the Michael line and the irrationals are discussed.