Maps of Ostaszewski and related spaces
An Ostaszewski space has come to mean a regular Hausdorff space of cardinality \(\aleph_1\) that is countably compact and in which every closed subset is countable or has countable complement. As unlikely as this combination of properties may at first seem, these spaces, since Ostaszewski's discovery, arise naturally in many situations. However their existence seems almost equivalent to \(\diamondsuit\) (Ostaszewski produced one from \(\diamondsuit\) and the author has, jointly with Eisworth, shown that even CH does not suffice). Nonetheless these are easily seen to be very germane to the topic of the present paper, the question: Is there a separable space \(X\), which (a) retracts onto every closed subspace, (b) is homeomorphic to every uncountable closed subset, (c) is homeomorphic to every uncountable continuous image (actually these were formulated in the dual category of Boolean algebras)? (Let us just mention the similarity to the `Toronto space' problem: Is there a non-discrete regular uncountable space which is homeomorphic to all of its uncountable subspaces?) While Shelah has, from \(\diamondsuit\), an example satisfying (a) and (b), and Bonnet and Rubin have (c), this paper investigates all of the combinations of the above question, examines more closely their axiomatic strengths and demonstrates that there are Cohen indestructible versions (thus, technically, exposing the lie to the reviewer's remark that the existence of these spaces may be equivalent to \(\diamondsuit\)). The paper closes with four interesting open problems.
scientific article; zbMATH DE number 937146