Scattered sets, chains and the Baire Category Theorem

Abstract

A scattered set of real numbers can be exhibited in terms of endpoints of the components of some nondecreasing sequence of open sets (chain). We study this connection between scattered sets and chains providing characterizations of scattered sets, semiscattered sets, certain splattered sets and sets with countable closure. It is shown that any application of the Baire Category Theorem on the real line leads naturally to a chain of open sets and hence to an exceptional scattered set. Some applications of this fact are given.