Spectral and T0-spaces in d-semantics

  • Authors:

  • Guram Bezhanishvili;Leo Esakia;David Gabelaia

  • Affiliations:

  • Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM;Department of Mathematical Logic, Razmadze Mathematical Institute, Tbilisi, Georgia;Department of Mathematical Logic, Razmadze Mathematical Institute, Tbilisi, Georgia

  • Venue:
  • TbiLLC'09 Proceedings of the 8th international tbilisi conference on Logic, language, and computation
  • Year:
  • 2009

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Abstract

In [6] it is shown that if we interpret modal diamond as the derived set operator of a topological space (the so-called d-semantics), then the modal logic of all topological spaces is wK4--weak K4--which is obtained by adding the weak version ⋄⋄p → p ∨ ⋄p of the K4-axiom ⋄⋄p → ⋄p to the basic modal logic K. In this paper we show that the T0 separation axiom is definable in d-semantics. We prove that the corresponding modal logic of T0-spaces, which is strictly in between wK4 and K4, has the finite model property and is the modal logic of all spectral spaces--an important class of spaces, which serve as duals of bounded distributive lattices. We also give a detailed proof that wK4 has the finite model property and is the modal logic of all topological spaces.