A non-normal topology generated by a two-point selection

We construct a two-point selection f : [ P ] 2 → P , where P is the set of the irrational numbers, such that the space ( P , τ f ) is not normal and it is not collectionwise Hausdorff either. Here, τ f denotes the topology generated by the two-point selection f. This example answers a question posed by V. Gutev and T. Nogura. We also show that if f : [ X ] 2 → X is a two-point selection such that the topology τ f has countable pseudocharacter, then τ f is a Tychonoff topology.