A non-normal topology generated by a two-point selection
For a set \(X\) and a two-point selection \(f:[X]^{2}\to X\), the topology \(\tau_{f}\) generated by \(f\) is Hausdorff and regular. This suggests the question whether or not \(\tau_{f}\) is a collectionwise Hausdorff topology on \(X\), or even a normal topology as asked by \textit{V. Gutev} and \textit{T. Nogura} [Topology Appl. 153, No. 5--6, 900--911 (2005; Zbl 1089.54005)]. In this paper, the authors construct a two-point selection \(f:[\mathbb{P}]^{2}\to\mathbb{P}\), where \(\mathbb{P}\) is the set of the irrational numbers, such that the topology \(\tau_{f}\) is not normal and not collectionwise Hausdorff either. It is also given that if \(f:[X]^{2}\to X\) is a two-point selection such that the topology \(\tau_{f}\) has countable pseudocharacter, then \(\tau_{f}\) is a Tychonoff topology.
scientific article; zbMATH DE number 5290941