In answer to a question of M. Reed, E. van Douwen and M. Wage [vDW79] constructed an example of a Moore space which had property D but was not pseudonormal. Their construction used the Martin’s Axiom type principle . We show that there is no such space in the usual Cohen model of the failure of CH.
We prove that implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality which has points . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.
Eric van Douwen produced in 1993 a maximal crowded extremally disconnected regular space and showed that its Stone-Čech compactification is an at most two-to-one image of . We prove that there are non-homeomorphic such images. We also develop some related properties of spaces which are absolute retracts of expanding on earlier work of Balcar and Błaszczyk (1990) and Simon (1987).
A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of . One especially important application, due to Veličković, was to the existence of nontrivial involutions on . A tie-point of has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set...
The cardinal invariants of are known to satisfy that . We prove that all inequalities can be strict. We also introduce a new upper bound for and show that it can be less than . The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).
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