A separable normal topological group which is not Lindelöf
An example mentioned in the title had been constructed under CH by \textit{A. Hajnal} and \textit{I. Juhász} [Gen. Topology Appl. 6, 199-205 (1976; Zbl 0323.22001)]; it was even hereditarily separable and hereditarily (collectionwise) normal. Now there is an example in ZFC, but without the hereditary properties. The construction is achieved by associating with each topological space X a semitopological group B(X) such that (a) X is a closed subspace of B(X), (b) the density is not increased, (c) B(X) is normal if all finite powers of X are normal, and (d) B(X) is a topological group if X is in addition sequentially compact (e.g.). Finally the authors show that the separable, normal, sequentially compact, non-compact space X of \textit{S. P. Franklin} and \textit{M. Rajagopalan} [Trans. Am. Math. Soc. 90, 305-314 (1971; Zbl 0217.481)] satisfies the condition in (c), and consequently B(X) is the desired example.
scientific article; zbMATH DE number 3922998