PP-2002-06:
Bezhanishvili, Guram and Gehrke, Mai
(2002)
*A New Proof of Completeness of S4 with respect to the Real Line.*
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## Abstract

It was proved in McKinsey and Tarski [7] that every finite

well-connected closure algebra is embedded into the closure algebra of

the power set of the real line R. Pucket [10] extended this result to

all finite connected closure algebras by showing that there exists an

open map from R to any finite connected topological space. We simplify

his proof considerably by using the correspondence between finite

topological spaces and finite quasi-ordered sets. As a consequence, we

obtain that the propositional modal system S4 of Lewis is complete

with respect to Boolean combinations of countable unions of convex

subsets of R, which is strengthening of McKinsey and Tarski's original

result. We also obtain that the propositional modal system Grz of

Grzegorczyk is complete with respect to Boolean combinations of open

subsets of R. Finally, we show that McKinsey and Tarski's result can

not be extended to countable connected closure algebras by proving

that no countable Alexandroff space containing an infinite ascending

chain is an open image of R.

Item Type: | Report |
---|---|

Report Nr: | PP-2002-06 |

Series Name: | Prepublication (PP) Series |

Year: | 2002 |

Uncontrolled Keywords: | modal logic, topological completeness, real line, Alexandroff space |

Subjects: | Logic |

Date Deposited: | 12 Oct 2016 14:36 |

Last Modified: | 12 Oct 2016 14:36 |

URI: | https://eprints.illc.uva.nl/id/eprint/70 |

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