Self-transversal spaces and their discrete subspaces
Two topologies \(\tau\) and \(\mu\) on a set \(X\) are said to be transversal if \(\tau\cup\mu\) is a subbase for the discrete topology on \(X\) . If there exists a bijection \(\varphi\) on \(X\) such that \(\tau\cup \{\varphi [U] : U\in\tau\}\) is a subbase for the discrete topology then the space \((X, \tau )\) is said to be self-transversal. This paper continues the study of self-transversal spaces which, in some way, was initiated by A. K. Steiner in the 60's and followed by some other authors since then. In this paper the authors answer some questions from a forthcoming paper by D. Shakhmatov, M. G. Tkachenko and R. G. Wilson; for instance, they show that under CH there is a compact scattered space which is not self-transversal and present a ZFC construction of compact self-transversal spaces either of arbitrarily large cardinality but with the Souslin property or without points of countable \(\pi\)-character. There are several results in the paper which are used to prove the main results like the ones commented before. Another nice result is that the fact that ``a metrizable space \(X\) is self-transversal if and only if its weight coincides with its cardinality'' is equivalent to the fact that there are only finitely many cardinals between \(\kappa\) and \(\kappa^\omega\) for every infinite cardinal \(\kappa\), and therefore consistent and independent of ZFC.
scientific article; zbMATH DE number 5001002