Concerning maximal sets
Gordon Thomas Whyburn · Bulletin of the American Mathematical Society · 1934
WHYBURN 1. Introduction.Let T be any monotone system of closed subsets of an arbitrary metric space M, that is, any closed subset of a set of the system T is itself a set of T. We shall call a set of the system T a T-set or a set of type T. We may also think of T as a property such that every closed subset of a set having this property likewise has this property.Since the null set (0) is a subset of every set, then, whatever be the system T, the null set is always a T-set.We first establish a general existence theorem for certain maximal sets relative to a system T. THEOREM.If N is any closed non-degenerate subset of a metric space M such that N is not disconnected by the omission of any T-set, then there exists a maximal subset (a continuum) H(N) of M containing N and having this property.