Metrizably fibered generalized ordered spaces (Q1399167)

Following \textit{V. V. Tkachuk} [Topol. Proc. 19, 321-324 (1994; Zbl 0854.54022)], a topological space \(X\) is metrizably fibered provided there is a continuous function \(f\) from \(X\) to a metric space \(M\) with the property that the fiber \(f^{-1}[m]\) of \(f\) is a metrizable subspace of \(X\) for each \(m\) in \(M\). A generalized ordered space (GO-space) is a Hausdorff space \(X\) equipped with a linear ordering such that \(X\) has a base of order-convex sets. The authors prove a characterization of metrizably fibered GO-spaces using certain quotient spaces and special open covers. They apply the characterization to deduce the following new result. Theorem: Let \(X\) be a perfect GO-space. Then \(X\) is metrizably fibered if and only if \(X\) has a \(\sigma\)-closed-discrete dense set. They also give several interesting examples of GO-spaces that are metrizably fibered as well as several other examples that are not metrizably fibered.