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

摘要

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.