The Big Bush machine

Abstract

In this paper we study an example-machine Bush(S,T) where S and T are disjoint dense subsets of R. We find some topological properties that Bush(S,T) always has, others that it never has, and still others that Bush(S,T) might or might not have, depending upon the choice of the disjoint dense sets S and T. For example, we show that every Bush(S,T) has a point-countable base, is hereditarily paracompact, is a non-Archimedean space, is monotonically ultra-paracompact, is almost base-compact, weakly α-favorable and a Baire space, and is an α-space in the sense of Hodel. We show that Bush(S,T) never has a σ-relatively discrete dense subset (and therefore cannot have a dense metrizable subspace), is never Lindelöf, and never has a σ-disjoint base, a σ-point-finite base, a quasi-development, a Gδ-diagonal, or a base of countable order. We show that Bush(S,T) cannot be a β-space in the sense of Hodel and cannot be a p-space in the sense of Arhangelskii or a Σ-space in the sense of Nagami. We show that Bush(P,Q) is not homeomorphic to Bush(Q,P). Finally, we show that a careful choice of the sets S and T can determine whether the space Bush(S,T) has strong completeness properties such as countable regular co-compactness, countable base compactness, countable subcompactness, and ω-Čech-completeness, and we use those results to find disjoint dense subsets S and T of R, each with cardinality 2ω, such that Bush(S,T) is not homeomorphic to Bush(T,S). We close with a family of questions for further study.