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

We construct a two-point selection , where P is the set of the irrational numbers, such that the space (P,τf) is not normal and it is not collectionwise Hausdorff either. Here, τf 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 is a two-point selection such that the topology τf has countable pseudocharacter, then τf is a Tychonoff topology.