The Big Bush machine

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