Judith Roitman

CH with no Ostaszewski spaces

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9947-99-02407-1. First published in Trans. Amer. Math. Soc. in 1999, published by the American Mathematical Society.

A space homeomorphic to each uncountable closed subspace under CH

Abstract Under CH there is a thin-tall locally compact scattered space which is homeomorphic to each uncountable closed subspace. This construction partially answera a Boolean Algebraic question of Matti Rubin.

Which spaces have a coarser connected Hausdorff topology

Abstract We present some answers to the title. For example, if K is compact, zero-dimensional and D is discrete, then K⊕D has a coarser connected topology iff w(K)⩽2|D|. Similar theorems hold for ordinal spaces and spaces K⊕D where K is compact, not necessarily zero-dimensional. Every infinite cardinal has a coarser connected Hausdorff topology; so do Kunen lines, Ostaszewski spaces, and Ψ-spaces; but spaces X with X⊂βω and |βω⧹X| c do not. The statement “every locally countable, locally compact extension of ω with cardinality ω1 has a coarser connected topology” is consistent with and independent of ZFC. If X is a Hausdorff space and w(X)⩽2κ, then X can be embedded in a Hausdorff space of density κ.

Maps of Ostaszewski and related spaces

Abstract Under ♦ there is an Ostaszewski space which is retractive, homeomorphic to every uncountable closed subspace, and homeomorphic to every locally countable regular Hausdorff uncountable continuous image, and also an Ostaszewski space with none of these properties. There is a ZFC example of a thin-tall locally compact scattered space for which each nonempty Cantor-Bendixson remainder is a retract. Attention is also paid to robustness under various types of forcing.

Uncountable autohomeomorphism groups of thin-tall locally compact scattered spaces

Under a hypothesis weaker thanCH, both the free group withω1 generators and the free Abelian group withω1 generators are non-trivial autohomeomorphism groups of locally compact thin-tall scattered spaces.

Autohomeomorphisms of Thin‐Tall Locally Compact Scattered Spaces in ZFC

ABSTRACT. For many uncountable groups G, there is a thin‐tall locally compact scattered space (equivalently, thin‐tall superatomic Boolean algebra) whose nontrivial autohomeomorphism (equivalently, automorphism) group is G. Such groups include ω!, the free group on ω1 generators, and the free commutative group on ω1.