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The Big Bush machine
| Content Provider | Semantic Scholar |
|---|---|
| Author | Bennett, Harold Burke, Dennis K. Lutzer, David |
| Copyright Year | 2012 |
| Abstract | 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 Lindelof, 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 -Cech-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. |
| Starting Page | 1514 |
| Ending Page | 1528 |
| Page Count | 15 |
| File Format | PDF HTM / HTML |
| DOI | 10.1016/j.topol.2010.07.037 |
| Volume Number | 159 |
| Alternate Webpage(s) | https://scholarworks.wm.edu/cgi/viewcontent.cgi?article=2588&context=aspubs |
| Alternate Webpage(s) | http://www.math.wm.edu/~lutzer/drafts/BigBushes.pdf |
| Alternate Webpage(s) | https://doi.org/10.1016/j.topol.2010.07.037 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
