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Erschienen in:
2011 | OriginalPaper | Buchkapitel
Spectral and T 0-Spaces in d-Semantics
verfasst von : Guram Bezhanishvili, Leo Esakia, David Gabelaia
Erschienen in: Logic, Language, and Computation
Verlag: Springer Berlin Heidelberg
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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
$\Diamond\Diamond p\to p\vee\Diamond p$
of the
K4
-axiom
$\Diamond\Diamond p\to\Diamond p$
to the basic modal logic
K
.
In this paper we show that the
T
0
separation axiom is definable in d-semantics. We prove that the corresponding modal logic of
T
0
-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.