The Big Bush Machine by

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 a 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 2, such that Bush(S, T ) is not homeomorphic to Bush(T, S). We close with a family of questions for further study.