We introduce a new combinatorial principle which we call ♣AD. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out de Caux type constructions of topological spaces. Our main result states that strong instances of ♣AD follow from the existence of a Souslin tree. It is also shown that the weakest instance of ♣AD does not follow from the existence of an almost Souslin tree. As an application, we obtain a simple, de Caux type proof of Rudin's result that if there is a Souslin tree, then there is an S-space which is Dowker.
Bibliographical noteFunding Information:
Some of the results of this paper come from the second author's M.Sc. thesis written under the supervision of the first author at Bar-Ilan University. We are grateful to Bill Weiss for kindly sharing with us a scan of Dahroug's handwritten notes with the construction of an Ostaszewski space from a Souslin tree and CH. Our thanks go to Tanmay Inamdar for many illuminating discussions, and to István Juhász for reading a preliminary version of this paper and providing a valuable feedback. We also thank the referee for a useful feedback. Both authors were partially supported by the Israel Science Foundation (grant agreement 2066/18). The first author was also partially supported by the European Research Council (grant agreement ERC-2018-StG 802756).
Both authors were partially supported by the Israel Science Foundation (grant agreement 2066/18 ). The first author was also partially supported by the European Research Council (grant agreement ERC-2018-StG 802756 ).
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- Almost disjoint
- Dowker space
- Ostaszewski space
- Souslin line