Ordered spaces all of whose continuous images are normal

Abstract

Abstract.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 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.

Countable successor ordinals as generalized ordered topological spaces

“…(2) Characterize Hausdorff spaces X such that all continuous images of X are normal. (This was partially solved by W. Fleissner and R. Levy in [5,6]). (3) Characterize Hausdorff spaces such that all continuous images of X are realcompact.…”

“…( 4) Characterize Hausdorff spaces such that all continuous images of X are paracompact. (5) Characterize Hausdorff spaces such that all continuous images of X are monotonically normal. (This is related to "Niekel Conjecture" answered positively by M. E. Rudin [12]).…”

Stone-Cech Remainders which Make Continuous Images Normal

“…Then Z is normal because it is the union of the normal space X and the compact space ClßX(Z\X). (See [FL,Lemma 1.1(c)]. ) Thus, if X is normal and ßX\X is finite, or more generally, discrete, then X is normality inducing, and hence ACRIN.…”