A guessing principle from a Souslin tree, with applications to topology

Assaf Rinot; Roy Shalev

doi:10.1016/j.topol.2022.108296

A guessing principle from a Souslin tree, with applications to topology

Assaf Rinot
, Roy Shalev

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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.

Original language	English
Article number	108296
Journal	Topology and its Applications
Volume	323
DOIs	https://doi.org/10.1016/j.topol.2022.108296
State	Published - 1 Jan 2023

Bibliographical note

Publisher Copyright:
© 2022 Elsevier B.V.

Funding

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 ).

Funders	Funder number
Bill Weiss
Horizon 2020 Framework Programme	802756
European Commission
Israel Science Foundation	2066/18

Keywords

Almost disjoint
Club
Dowker space
Ostaszewski space
Souslin line

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10.1016/j.topol.2022.108296

Cite this

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AB - 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.

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KW - Ostaszewski space

KW - Souslin line

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A guessing principle from a Souslin tree, with applications to topology

Abstract

Bibliographical note

Funding

Keywords

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