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Trees and gaps from a construction scheme


Authors: Fulgencio Lopez and Stevo Todorcevic
Journal: Proc. Amer. Math. Soc. 145 (2017), 871-879
MSC (2010): Primary 03E05, 03E35, 03E65
DOI: https://doi.org/10.1090/proc/13431
Published electronically: November 3, 2016
MathSciNet review: 3577886
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Abstract: We present simple constructions of trees and gaps using a general construction scheme that can be useful in constructing many other structures. As a result, we solve a natural problem about Hausdorff gaps in the quotient algebra $ \mathcal {P}(\omega )/{\rm Fin}$ found in the literature. As it is well known, Hausdorff gaps can sometimes be filled in $ \omega _1$-preserving forcing extensions. There are two natural conditions on Hausdorff gaps, dubbed $ S$ and $ T$ in the literature, that guarantee the existence of such forcing extensions. In part, these conditions are motivated by analogies between fillable Hausdorff gaps and Suslin trees. While the condition $ S$ is equivalent to the existence of $ \omega _1$-preserving forcing extensions that fill the gap, we show here that its natural strengthening $ T$ is in fact strictly stronger.


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Additional Information

Fulgencio Lopez
Affiliation: Department of Mathematics, University of Toronto, Bahen Center, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
Email: fulgencio.lopez@mail.utoronto.ca

Stevo Todorcevic
Affiliation: Department of Mathematics, University of Toronto, Bahen Center, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4; Institut de Mathématiques de Jussieu, UMR 7586, 2 pl. Jussieu, case 7012, 75251 Paris Cedex 05, France
Email: stevo@math.toronto.edu, stevo.todorcevic@imj-prg.fr

DOI: https://doi.org/10.1090/proc/13431
Keywords: Construction schemes, Suslin tree, destructible gaps, S-gaps, T-gaps.
Received by editor(s): July 13, 2015
Received by editor(s) in revised form: April 3, 2016
Published electronically: November 3, 2016
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2016 American Mathematical Society

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