Images of the countable ordinals (Q2400897)

Spaces that are continuous images of the standard space \([0,\omega_1)\) of the countable ordinals are investigated. First the authors observe that if \(Y\) is such a space satisfying the \(T_3\)-axiom, then \(Y\) has a monotonically normal compactification, and is monotonically normal, locally compact and scattered. Examples illustrate that the condition of regularity is necessary for these results. They then study when a regular continuous image of the countable ordinals is compact, paracompact, or metrizable. For instance they prove that metrizability of such a \(Y\) is equivalent to each of the following conditions: \(Y\) has a \(G_\delta\)-diagonal, \(Y\) is perfect, \(Y\) has a point-countable base, \(Y\) has a small diagonal (in the sense of Hušek), and \(Y\) has a \(\sigma\)-minimal base. As part of their investigations they also discuss the (historical) connections between their result that any compact Hausdorff space \(Y\) with \(| Y| \leq \aleph_1\) and possessing a small diagonal is metrizable and a recent theorem by Gruenhage that each scattered compact Hausdorff space with a small diagonal is metrizable. In particular they show how Gruenhage's result can also be derived with the help of older work of \textit{S. Mrowka} et al. [Lect. Notes Math. 378, 288--297 (1974; Zbl 0299.54015)]. The paper contains a lot of background information and the authors clearly tried to make it accessible to beginners. By the convention of the authors, a space is a \(T_1\)-topological space.