Ordered spaces all of whose continuous images are normal
Proceedings of the American Mathematical Society
Some spaces, such as compact Hausdorff spaces, have the property that every regular continuous image is normal. In this paper, we look at such spaces. In particular, it is shown that if a normal space has finite Stone-Cech remainder, then every continuous image is normal. A consequence is that every continuous image of a Dedekind complete linearly ordered topological space of uncountable cofinality and coinitiality is normal. The normality of continuous images of other ordered spaces is also
... d spaces is also discussed. Let A' be a normal topological space. In general, there is no reason to expect every regular continuous image of X to be normal. After all, every space, normal or not, is the continuous image of a normal space, namely, a discrete space. On the other hand, every continuous image of certain spaces is normal. For example, if X is compact Hausdorff, or, more generally, regular Lindelöf, then every (regular) continuous image of X is normal, because every continuous image of X is Lindelöf. In this note, we discuss spaces with the property that every regular continuous image is normal. In particular, we point out that those metric spaces which have this property are the separable ones, and, we discuss the situation for complete linearly ordered topological spaces. We make the convention that all given spaces are assumed to be regular, which includes Hausdorff. All metric spaces will be assumed to have the metric topology. All linearly ordered spaces will be assumed to have the order topology. Ordinals will be von Neumann ordinals and cardinals will be initial ordinals.