Spectral and T0-Spaces in d-Semantics

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.