On modal logics arising from scattered locally compact Hausdorff spaces

On Modal Logics Arising from Scattered Locally Compact Hausdorff Spaces

For a topological space X, let L(X) be the modal logic of X where is interpreted as interior (and hence ♦ as closure) in X. It was shown in [6] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, S4.Grzn (n ≥ 1), 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 ...

Modal compact Hausdorff spaces

We introduce modal compact Hausdorff spaces as generalizations of modal spaces, and show these are coalgebras for the Vietoris functor on compact Hausdorff spaces. Modal compact regular frames and modal de Vries algebras are introduced as algebraic counterparts of modal compact Hausdorff spaces, and dualities are given for the categories involved. These extend the familiar Isbell and de Vries d...

Topology Proceedings UNIVERSAL LOCALLY COMPACT SCATTERED SPACES

If δ is an ordinal, we denote by C(δ) the class of all cardinal sequences of length δ of locally compact scattered (in short: LCS) spaces. If λ is an infinite cardinal, we write Cλ(δ) = {s ∈ C(δ) : s(0) = λ = min[s(ζ) : ζ < δ]}. An LCS space X is called Cλ(δ)-universal if SEQ(X) ∈ Cλ(δ), and for each sequence s ∈ Cλ(δ) there is an open subspace Y of X with SEQ(Y ) = s. We show that • there is a...

Logics for Compact Hausdorff Spaces via de Vries Duality

In this thesis, we introduce a finitary logic which is sound and complete with respect to de Vries algebras, and hence by de Vries duality, this logic can be regarded as logic of compact Hausdorff spaces. In order to achieve this, we first introduce a system S which is sound and complete with respect to a wider class of algebras. We will also define Π2-rules and establish a connection between Π...

ALBA-style Sahlqvist preservation for modal compact Hausdorff spaces