Self-transversal spaces and their discrete subspaces

Abstract

A space X is called self-transversal if there is a bijection φ : X → X such that the family τ(X) ⋃ φ(τ(X)) forms a subbase of the discrete topology on X. We prove that, under CH, there exists a compact scattered space which is not self-transversal. It is shown that there exist compact self-transversal spaces of arbitrarily large cardinality with the Souslin property. We present examples of compact spaces which give a negative answer in ZFC to Problems 2 and 3 from [8] and a partial negative answer to Problem 1 of [8]. We also establish that it is independent of ZFC whether any metrizable space X is self-transversal if and only if w(X) = |X|. We show that any monotonically normal scattered space is selftransversal and that adding a single point to a self-transversal space can destroy self-transversality. © 2005 Rocky Mountain Mathematics Consortium.