PP-2017-14:
Bezhanishvili, Guram and Bezhanishvili, Nick and Lucero-Bryan, Joel and van Mill, Jan
(2017)
*On modal logics arising from scattered locally compact Hausdorff spaces.*
[Pre-print]

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## Abstract

For a topological space X, let L(X) be the modal logic of X where \Box is interpreted as interior (and hence \Diamond as closure) in X. It is known that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, S4.Grz_n, and their intersections arise as L(X) for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [6, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or S4.Grz_n for some n \geq 1. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.

Item Type: | Pre-print |
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Report Nr: | PP-2017-14 |

Series Name: | Prepublication (PP) Series |

Year: | 2017 |

Subjects: | Logic Mathematics |

Depositing User: | Nick Bezhanishvili |

Date Deposited: | 15 Oct 2017 13:56 |

Last Modified: | 19 Oct 2017 16:11 |

URI: | https://eprints.illc.uva.nl/id/eprint/1566 |

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