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Mckinsey Algebras and Topological Models of S4.1

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Mckinsey Algebras and Topological Models of S4.1 McKinsey Algebras and Topological Models of S4.1 Thomas Mormann Abstract. The aim of this paper is to show that every topological space gives rise to a wealth of topological models of the modal logic S4.1. The construction of these models is based on the fact that every topological space defines a Boolean closure algebra (to be called a McKinsey algebra) that neatly reflects the structure of the modal system S4.1 in that all its elements satisfy the topological interpretation of the McKinsey axiom. It is shown that the class of topological models based on McKinsey algebras contains a canonical model that can be used to prove a completeness theorem for S4.1. Finally, it is proved that the McKinsey algebra MKX of a space X endowed with an α-topology satisfies Esakia’s GRZ axiom. Keywords. McKinsey Axiom Schema, Modal Logic, Topological Models of S4, S4.1, Boun- dary, Closure Algebra, Completeness. 1. Introduction. Let S4.1 be the propositional modal logic over S4 defined by adding the so called McKinsey axiom (schema) (1.1) ϕ → ϕ to the axioms and rules of S4. Let (X, τ) be a topological space with closure operator cl and interior operator int (cf. Kuratowski and Mostowski (1976), Willard (2004)). Inter- preting the modal operators and by the topological operators int and cl in the familiar way, the McKinsey axiom defines the following subset MKX of the power set PX of X: (1.2) MKX := {a; intcl(a) ⊆ clint(a), a ⊆ X} In view of (1.1) I propose to call the elements of MKX as McKinsey sets. The power set PX of a topological space X has the structure of a complete interior Boolean algebra (PX, ⊆, Ø, X, int) or, equivalently, the structure of a complete closure algebra (PX, ⊆, Ø, X, cl). Throughout this paper PX is assumed to carry this structure of a modal Boolean algebra. It is natural to ask then how the subset MKX of PX behaves with respect to this structure. The main technical result of this paper is the following theorem: (1.3) Theorem. Let (X, τ) be a topological space. The set MKX is a Boolean subalgebra of the Boolean algebra PX. More precisely, MKX is a modal subalgebra of PX. The modal algebra MKX is to be called the McKinsey algebra of the topological space X. The elements of MKX are referred to as McKinsey sets.♦1 The following remarks on (1.3) may be in order: (i) Let OX and CX denote the sets of open and of closed sets of a topological space (X, τ), respectively. Then OX ∪ CX ⊆ MKX: For a ∈ OX this is proved as follows. From a = int(a) one obtains and intcl(a) = intclint(a) ⊆ clint(a) by the fundamental properties of the interior operator int. Hence a ∈ MKX. The assertion for CX is proved similarly. As will be shown later MKX not only contains open and closed sets but also all kinds of „mixed“ sets, in particular finite intersections and unions of open and closed sets and nowhere dense sets. In a sense MKX may be considered as a „finite“ counterpart or analogue of the σ-complete Boolean algebra BOX of Borel sets of X. This is nicely evidenced by Theorem (2.4), to be proved in section 4 according to which certain quotients of MKX and BOX, respectively, are isomorphic Boolean algebras. Nevertheless MKX and BOX are clearly different: Neither MKX is a subset of BOX, nor is BOX a subset of MKX. (ii) A particularly important role for the structure of MKX is played by the ideal NDX of nowhere dense sets. Recall that a subset a of a topological space is called nowhere dense if and only if intcl(a) = Ø. One immediately observes that NDX ⊆ MKX since by definition one has Ø = intcl a ⊆ clint a = Ø. As will be shown later, OX and NDX generate MKX as a Boolean algebra in a precise sense. 1 Stone (1937) seems to have been the first paper where MKX showed up (not under this name of course). Stone called it the „complete basic ring of the topological space X“ and proved some of ist the basic properties of McKinsey sets (cf. Theorem 24). Much later, Esakia (2004) mentions as a „key observation“ that the subset {a ∈ B; intcla ≤ clinta} of a Boolean closure algebra B is also a Boolean closure algebra (cf. Esakia (2004, section 8.4). He explicitly mentioned the McKinsey axiom but seems to be unaware of the fact that already Stone made the just mentioned „key observation“. Independently of Stone, McKinsey, and Esakia, the set MKX was mentioned by several topologists who dubbed the „McKinsey sets“ of this paper „delta sets“, „NDB sets (= sets with nowhere dense boundary), or „α-open sets“. These authors, however, never noticed that these sets played a role in modal logic, nor realized that MKX is a Boolean algebra. 2 (iii) In general MKX is a proper subalgebra of PX as is shown by the following canonical example: Consider the real line R endowed with its standard Euclidean topology. Then one obtains for the set Q of rational numbers (1.4) intcl(Q) = R and clint(Q) = Ø. Hence Q is not an element of MKR. This proves that in general MKX ≠ PX. Since every singleton {r}, r ∈ R, is clearly a McKinsey set, the set Q as a countable union of rational numbers evidences that MKX in general is not a σ-complete Boolean algebra. On the other hand it is well known that there are topological spaces X for which PX = MKX, i.e., every subset a of X is a McKinsey set (cf. Bezhanishvili, Esakia, Gabelaia (2005), van Benthem, Bezhanishvili (2007, Chapter 5, p. 253).2 The main aim of this paper is to use the modal algebra MKX to define a profusion of sound topological models of S4.1 and related logics. Among these models there is also a canonical complete model that allows us to prove a completeness theorem for S4.1. The outline of this paper is as follows. In the next section we explore the lattice-theoretical context in which MKX is embedded. That is to say, we study in some detail the various topologically defined lattices to which MKX is related. This prepares the ground for proving (1.2) in the following section, exploiting a formula of A.H. Stone mentioned. Then we show that McKinsey algebras may be used to construct sound topological models of S4.1. Among these topological models for S4.1 there is a canonical model endowed with a canonical interpretation. This enables us to prove a completeness for S4.1 following the lines of the elegant completeness proof for S4 presented by Aiello, van Benthem, Bezha- nishvili (2003)). 2. McKinsey algebras and other lattices of topological spaces. The algebra MKX is in no way the only lattice that is related with a topological space (X, τ). On the contrary, MKX may be considered as an almost completely neglected member of an extended family of lattice structures that can be associated to any topological space X. For later use, some of them are collected in the following list: 2 Bezhanishvili, Esakia, Gabelaia (2005) proposed to call a topological space X a McKinsey space if the set of dense subsets of X forms a filter (ibid., p. 326). They showed that for McKinsey spaces the algebras MKX and PX coincide, i.e., all subsets of a X satisfy the requirement intcl(a) ⊆ clint(a). Hence calling MKX the McKinsey algebra of X is quite in line with this usage. 3 (2.1) Definition. Let (X, τ) be a topological space. The topological structure on X gives rise to (at least) the following lattices: (1) PX := modal Boolean algebra of subsets of X. (2) OX = Heyting algebra of open sets. (3) CX = Co-Heyting algebra of closed sets. (4) ROX = Boolean algebra of regular open sets. (5) RCX = Boolean algebra of regular closed sets. (6) CLX = Boolean algebra of clopen sets. (7) BOX = Boolean algebra of Borel sets. (8) MKX = modal Boolean algebra of McKinsey sets (with closure operator cl).♦ The general aim of this paper is to show that for virtually all considerations of modal logic the modal algebra (PX, ⊆, cl) can be replaced by the smaller modal algebra (MKX, ⊆, cl). As an extra bonus of this replacement one obtains an easy completeness theorem for S4.1 instead of S4. The following inclusion relations between these lattices listed in (2.1) are well-known: (2.2) Proposition: Let (X, τ) be a topological space. Then the following inclusion relations (as inclusions of order structures) between the lattices defined above hold. ROX ⊆ OX MKX CLOX ⊆ ⊆ ⊆ PX.♦ RCX ⊆ CX BOX3 Recall that a subset a of a topological space X is meager if and only if a is the countable union of nowhere dense sets. The set of meager sets is denoted by MGX. MGX is well- known to be an ideal of BOX. In Halmos (1963) the following somewhat surprising relation between the Boolean algebras ROX, and BOX is proved (cf. Halmos 1963, § 13, Theorem 4, p. 58): 3 In general, MKX is not a subset of BOX, nor is BOX a subset of MKX. Already in (1.3) it was pointed out that the set Q of rational numbers is a Borel set but not a McKinsey set. In order to show that in general MKX is not a subset of BOX one may argue as follows. As is well-known the Cantor set C is a closed nowhere dense set of R.
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