aa r X i v : . [ m a t h . L O ] F e b THE BAIRE CLOSURE AND ITS LOGIC
G. BEZHANISHVILI AND D. FERNANDEZ-DUQUE
Abstract.
The Baire algebra of a topological space X is the quotient of thealgebra of all subsets of X modulo the meager sets. We show that this Booleanalgebra can be endowed with a natural closure operator satisfying the standardproperties of the topological one, resulting in a closure algebra which we denote Baire ( X ). We identify the modal logic of such algebras to be the well-knownsystem S5 , and prove soundness and strong completeness for the cases where X is crowded and either completely metrizable and continuum-sized or locallycompact Hausdorff. We also show that every extension of S5 is the modal logicof a subalgebra of Baire ( X ), and that soundness and strong completeness alsoholds in the language with the universal modality. Introduction
Among the canonical examples of Boolean algebras are the powerset ℘ ( X ) of aset X , as well as its subalgebras; indeed, every Boolean algebra is isomorphic toa subalgebra of a powerset algebra (see, e.g., [18, p. 28, Thm. 2.1]). Alternately,one can consider quotients of the form ℘ ( X ) / I , where I is a suitable ideal of ℘ ( X ).Some of the most familiar such ideals are the ideal N of null sets when X is ameasure space (see, e.g., [18, p. 233, Exmp. 14.27]), or the ideal M of meager setswhen X is a topological space (see, e.g., [18, p. 182, Lem. 12.8]); recall that a setis nowhere dense if the interior of its closure is empty, and that it is meager if it isa countable union of nowhere dense sets. The quotient ℘ ( X ) / M gives rise to the Baire algebra of X .In the case that X is a topological space, the powerset algebra of X comesequipped with the usual closure operator c : ℘ ( X ) → ℘ ( X ). This operator satisfiessome familiar properties also known as the Kuratowski axioms, including e.g. A ⊆ c A (see Section 2 for the full list), and any operator satisfying these axioms uniquelydetermines a topology on X . One can more generally consider a Boolean algebra B with an operator c : B → B satisfying the same axioms; such algebras are the closure algebras of McKinsey and Tarski [23]. It is then a natural question to askwhether a closure operator on ℘ ( X ) carries over to quotients ℘ ( X ) / I in a meaningfulway. This question has already been answered in the affirmative by Fern´andez-Duque [11] and Lando [21] in the setting of the Lebesgue measure algebra, definedas the quotient of the Borel sets of reals modulo the null sets. In this article, wewill explore this question in the setting of the Baire algebra of a topological space X .One subtlety when defining a closure operator for such quotients is that, denotingthe equivalence class of Y by [ Y ], the definition c [ Y ] := [ c Y ] does not yield a well-defined operation: for example, in the above-mentioned quotients [ Q ] = [ ∅ ] (as Q Mathematics Subject Classification.
Key words and phrases.
Modal logic; topological semantics; Baire space; resolvable space. is both meager and of measure zero) yet [ c Q ] = [ R ] = [ ∅ ] = [ c ∅ ]. Instead, we mustcompute the closure of a set directly within the quotient, by the expression c a = inf { [ C ] : a ⊑ [ C ] and C is closed } , where ⊑ denotes the partial order on the quotient algebra. We utilize the BanachCategory Theorem (see, e.g., [24, Thm. 16.1]) to show that this produces a clo-sure operator on the Baire algebra of any topological space X , and we denote theresulting closure algebra by Baire ( X ).Much as Boolean algebras provide semantics for propositional logic, closure al-gebras provide semantics for modal logic, which extends propositional logic withan operator ♦ that we will interpret as a closure operator, along with its dual (cid:3) ,interpreted as interior. Such topological semantics of modal logic, as well as theclosely related intuitionistic logic, predates their now-widespread relational seman-tics. For intuitionistic logic it was first developed by Stone [28] and Tarski [29], andfor modal logic by Tsao-Chen [30], McKinsey [22], and McKinsey and Tarski [23].Under this interpretation, the modal logic of all topological spaces turns out to bethe well-known modal system of Lewis, S4 (see Section 2 for the definition). Otherwell-known extensions of S4 also turn out to be the modal logics of interestingtopological spaces. To give a couple of examples: • S4 . := S4 + ♦(cid:3) p → (cid:3)♦ p is the modal logic of all extremally disconnectedspaces [12]. • S4 . := S4 + (cid:3) ( (cid:3) p → q ) ∨ (cid:3) ( (cid:3) q → p ) is the modal logic of all hereditarilyextremally disconnected spaces [1].Here we recall that a space X is extremally disconnected if the closure of each openset is open, and X is hereditarily extremally disconnected if each subspace of X isextremally disconnected.These topological completeness results can be strengthened as follows. By thecelebrated McKinsey-Tarski theorem [23], S4 is the modal logic of any crowdedmetrizable space. In fact, S4 is strongly sound and complete with respect to anycrowded metrizable space [13, 19]. In particular, S4 is the modal logic of the realunit interval [0 , S4 . is the modal logic of the Gleasoncover of [0 ,
1] (see [4]), while S4 . is the modal logic of a countable (hereditarily)extremally disconnected subspace of the Gleason cover of [0 ,
1] (see [1]). In contrast, S5 := S4 + ♦ p → (cid:3)♦ p , which is one of the best known modal logics, is not completewith respect to any class of spaces that satisfy even weak separation axioms. Indeed, ♦(cid:3) p → (cid:3) p is valid in a topological space iff each open set is also closed. Thus, a T -space validates S5 iff it is discrete.One can also characterize the modal logics of closure algebras that are not basedon a powerset. Fern´andez-Duque [11] and Lando [21] have shown that S4 is the logicof the Lebesgue measure algebra, and in this article we will characterize the logic ofBaire algebras. An element c ∈ Baire ( X ) is closed (resp. open) if c = [ C ] for someclosed (resp. open) C . A distinguished feature of Baire ( X ) is that c ∈ Baire ( X )is open iff it is closed. This yields that S5 is sound with respect to Baire ( X )for any topological space X . For completeness, we refine Hewitt’s [17] well-known We recall that a space X is crowded or dense-in-itself if it has no isolated points. It is worthpointing out that the original McKinsey-Tarski theorem also had the separability (equivalentlythe second countability) assumption on X , which was later removed by Rasiowa and Sikorski [25]by an elaborate use of the Axiom of Choice. For a modern proof of this result see [3]. HE BAIRE CLOSURE AND ITS LOGIC 3 concept of resolvability to that of
Baire resolvability.
A space is resolvable if itcan be partitioned into two dense sets. Similarly, a closure algebra B is resolvableif there are a, b that are orthogonal ( a ∧ b = 0) and dense ( c a = c b = 1). If wedenote | R | by c , we show that if X is crowded and either a complete metric spaceof cardinality c or a locally compact Hausdorff space, then Baire ( X ) is resolvable.In fact, such algebras are c -resolvable, meaning that we can find c -many dense andpairwise orthogonal elements of Baire ( X ). Our main tool in proving these resultsis the Disjoint Refinement Lemma (see, e.g., [9, Lem. 7.5]).Using these resolvability results we show that if X is a crowded, continuum-sized,complete metrizable space, then S5 is strongly complete for Baire ( X ), yielding avariant of the McKinsey-Tarski theorem for S5 . In view of c -resolvability, strongcompleteness holds even if we extend the propositional language with continuum-many propositional variables. This yields a sharper version of strong completenessthan that given in the literature, where only countable languages are typicallyconsidered. Some of our other main results include that every extension of S5 isthe modal logic of some subalgebra of Baire ( X ) for any crowded second-countablecompletely metrizable space X . These results also hold if we replace completelymetrizable by locally compact Hausdorff. Finally, we show how to extend our resultsto the setting of the universal modality.2. Preliminaries
We assume some basic familiarity with Boolean algebras, topological spaces, andordinal and cardinal arithmetic (see, e.g., [18, 10, 9]). In this section we will reviewclosure algebras and their relation to modal logic.2.1.
Closure algebras.
For each topological space X , the powerset ℘ ( X ) is aBoolean algebra and the closure operator c : ℘ ( X ) → ℘ ( X ) satisfies the Kuratowskiaxioms: A ⊆ c A, cc A ⊆ c A, c ( A ∪ B ) = c A ∪ c B, and c ∅ = ∅ . We call the pair ( ℘ ( X ) , c ) the Kuratowski algebra of X and denote it by Kur ( X ).McKinsey and Tarski [23] generalized the concept of a closure on ℘ ( X ) to thatof a closure on an arbitrary Boolean algebra: Definition 2.1. A closure algebra is a pair ( B, c ), where B is a Boolean algebraand c : B → B satisfies, for every a, b ∈ B ,(1) a ≤ c a ,(2) cc a ≤ c a ,(3) c ( a ∨ b ) = c a ∨ c b, and(4) c i a := − c − a . If c moreover satisfies c a = ic a, then ( B, c ) is a monadic algebra. The notion of monadic algebras is due to Halmos [15]. As is the case for Booleanalgebras, closure algebras can be represented as subalgebras of algebras based on apowerset.
Theorem 2.2. [23, Thm. 2.4]
Each closure algebra is isomorphic to a subalgebraof the Kuratowski algebra
Kur ( X ) of some topological space X . G. BEZHANISHVILI AND D. FERNANDEZ-DUQUE
Recall that Boolean algebras are partial orders ( B, ≤ ) with a least and greatestelement, usually denoted 0 and 1, and where for a, b ∈ B their meet (greatest lowerbound) a ∧ b and complement − a are defined, and satisfy the usual axioms (see,e.g., [16, 27]). If S ⊆ B , then V S denotes its meet and W S its join when theyexist; the algebra B is complete provided these always exist. The next definitiongoes back to Halmos [15]. Definition 2.3.
We say that A ⊆ B is relatively complete in B if for each b ∈ B ,the set { a ∈ A | b ≤ a } has a least element (which then is V { a ∈ A | b ≤ a } ).Of particular importance are countable joins and meets, also called σ -joins and σ -meets; the algebra B is σ -complete if such joins and meets always exist. An ideal is a set I ⊆ B which is closed under finite joins and which for each element a ∈ I also contains all elements underneath a ; it is proper if 1 / ∈ I ; and it is a σ -ideal if it is closed under σ -joins. If I is an ideal, then the quotient B/ I (defined inthe standard way) is a Boolean algebra; moreover, if I is a σ -ideal, then B/ I is σ -complete (see, e.g., [16]).2.2. Modal logic.
We will work with the basic unimodal language as well as itsextension with the universal modality. However, as we are interested in strongcompleteness for possibly uncountable sets of formulas, we want to allow for anarbitrary number of propositional variables. Given a cardinal λ , let P λ = { p ι | ι <λ } be a set of λ -many propositional variables. The modal language L λ ♦ ∀ is definedby the grammar (in Backus-Naur form) ϕ, ψ := p | ϕ ∧ ψ | ¬ ϕ | ♦ ϕ | ∀ ϕ where p ∈ P λ . We also use standard shorthands, for example defining ⊥ as p ∧ ¬ p (where p is any fixed variable), ϕ ∨ ψ as ¬ ( ¬ ϕ ∧ ¬ ψ ), and (cid:3) ϕ as ¬ ♦ ¬ ϕ . We denotethe ∀ -free fragment by L λ ♦ , and we omit the superscript λ when λ = ω .We use ♦ as primitive rather than (cid:3) since, historically, algebraic semantics waspresented in terms of closure-like operators on Boolean algebras. We restrict ourattention to logics above S4 , and especially to S5 and its extensions. A standardaxiomatization of S4 in L λ ♦ is given by all (classical) propositional tautologies andthe axioms and rulesM : ♦ ( p ∨ q ) → ♦ p ∨ ♦ q T : p → ♦ p : ♦♦ p → ♦ p N : ¬ ♦ ⊥ Sub : ϕ ( p , . . . , p n ) ϕ ( ψ , . . . , ψ n )MP : ϕ ϕ → ψψ Mon : ϕ → ψ ♦ ϕ → ♦ ψ .The logic S5 is then obtained by adding the axiom 5: ♦ p → (cid:3)♦ p . Given a logicΛ over L λ ♦ , we obtain a new logic Λ U by adding the S5 axioms and rules for ∀ (orrather for ∃ := ¬∀¬ ) and the connecting axiom ♦ ϕ → ∃ ϕ . We will specifically beinterested in S5U .The language L λ ♦ has familiar Kripke semantics based on frames F = ( W, R ),where R ⊆ W × W (see, e.g., [7, 6]). We will not review this semantics in detail, HE BAIRE CLOSURE AND ITS LOGIC 5 and instead regard it as a special case of topological or, more generally, algebraicsemantics.Closure algebras provide the algebraic semantics for S4 and its normal extensions,meaning those logics containing the axioms of S4 and closed under the rules of S4 . Definition 2.4. An algebraic model of S4U is a structure M = ( B, J · K ) , where B isa closure algebra and J · K : L λ ♦ ∀ → B is a valuation for L λ ♦ ∀ ; that is, a function suchthat J p K ∈ B for each p ∈ P λ and J ϕ ∧ ψ K = J ϕ K ∧ J ψ K J ¬ ϕ K = − J ϕ KJ ♦ ϕ K = c J ϕ K J ∀ ϕ K = ( J ϕ K = 10 otherwise.We may also say that M is an algebraic model of S4 , and that the restriction of J · K to L λ ♦ is a valuation for L λ ♦ . Validity, soundness, and completeness are then defined in the usual way:
Definition 2.5.
Let M = ( B, J · K ) be an algebraic model and Γ ∪ { ϕ } ⊆ L λ ♦ ∀ .(1) We say that ϕ is valid in M , written M | = ϕ , if J ϕ K = 1. If M | = γ for each γ ∈ Γ, then we say that Γ is valid in M and write M | = Γ.(2) We write B | = ϕ if ( B, J · K ) | = ϕ for every valuation J · K on B , and define B | = Γ similarly.(3) If Ω is a class of closure algebras, we say that ϕ is valid in Ω provided B | = ϕ for each B ∈ Ω. That Γ is valid in
Ω is defined similarly.(4) For S ⊆ { ♦ , ∀} , we denote the set of valid L λS -formulas in Ω by Log S (Ω). IfΩ = { B } , we simply write Log S ( B ).(5) A logic Λ is sound for Ω if Λ ⊆ Log S (Ω), and complete if Λ ⊇ Log S (Ω).We will also be interested in strong completeness of logics. Definition 2.6.
Let ϕ be a formula, Γ a set of formulas, and Λ a logic in L λ ♦ ∀ .(1) We write Γ ⊢ Λ ϕ if ϕ is derivable from Γ in Λ, treating elements of Γ asaxioms. We omit Γ when Γ = ∅ .(2) For a closure algebra B we write Γ | = B ϕ if B | = Γ implies that B | = ϕ .(3) We say that Λ is strongly complete for a closure algebra B if Γ | = B ϕ impliesΓ ⊢ Λ ϕ .We remark that strong completeness, as we defined it, does not imply soundness.As we had mentioned previously, Kripke frames can be seen as a special case ofKuratowski algebras. If W is a set and R ⊆ W × W , then R induces an operatoron ℘ ( W ) given by A R − ( A ) := { w ∈ W | wRa for some a ∈ A } . If R is reflexive and transitive, we say that ( W, R ) is an S4 -frame . It is well knownand not hard to check that in this case ( ℘ ( W ) , R − ) is a Kuratowski algebra, andthat it is the Kuratowski algebra of the topology on W whose open sets are those U ⊆ W that satisfy R ( U ) ⊆ U , recalling that R ( U ) = { w ∈ W | uRw for some u ∈ U } . If in addition R is symmetric, then we say that ( W, R ) is an S5 -frame. We willtacitly identify an S4 -frame with the associated topological space and even withthe associated Kuratowski algebra. G. BEZHANISHVILI AND D. FERNANDEZ-DUQUE
It is well known that the topologies arising from S4 -frames are Alexandroff spaces ;that is, topologies where arbitrary intersections of open sets are open. It is aconsequence of the results of McKinsey and Tarski [23] and Kripke [20] that S4 is sound for the class of closure algebras, and complete for the class of S4 -frames.It is well known that S4 and S5 are strongly complete, both with respect to theconsequence relation we are considering (also known as the ‘global consequencerelation’) [25] and the ‘local’ consequence relation, which is defined pointwise [6].The literature typically considers countable languages, but the adaptation of theseresults to uncountable languages is straightforward; we provide a sketch below. Theorem 2.7.
For an infinite cardinal λ , we have: (1) S4 λ and S4U λ are strongly complete for the class of all S4 -frames of cardi-nality at most λ . (2) S5 λ and S5U λ are strongly complete for the class of all S5 -frames of cardi-nality at most λ .Proof sketch. We outline how a standard proof as given e.g. in [6] can be adaptedto obtain the results as stated. Let Λ be one of the logics mentioned above, andsuppose that Γ Λ ϕ . The completeness proof by the relativized canonical model M Γ c = ( W Γ c , R Γ c , J · K Γ c ), where W Γ c is the set of maximal consistent sets containing Γ,carries over mostly verbatim for uncountable languages. The relativized canonicalmodel has the property that M Γ c | = Γ and M Γ c = ϕ . The only caveat is thatthe proof of the Lindenbaum lemma requires care. This lemma states that anyconsistent set of formulas ∆ can be extended to a maximal consistent set ∆ ′ ⊇ ∆.However, this is readily proven in the general case using Zorn’s lemma.The set of worlds W Γ c may have cardinality greater than λ . However, viewing L λ ♦ ∀ as a fragment of first order logic, we may apply the downward L¨owenheim-Skolem theorem to obtain a model M of size at most λ , such that M | = Γ and M = ϕ , as required. (cid:3) For a topological space X , it is well known (and not hard to see) that thefollowing are equivalent:(1) X | = S5 ;(2) every open set of X is closed;(3) X has a basis which is also a partition of X ;(4) X is an S5 -frame.Thus, unlike the case of S4 , no true generality is obtained by passing to topologicalsemantics of S5 . As we will see, the story is quite different for algebraic semantics.We conclude this preliminary section by recalling the extensions of S5 . For eachnatural number n ≥
1, letAlt n := n +1 ^ i =1 ♦ p i → _ ≤ i Theorem 2.8 (Scrogg’s Theorem) . The consistent extensions of S5 form the fol-lowing ( ω + 1) -chain ( with respect to ⊃ ) : S5 ⊃ · · · ⊃ S5 n ⊃ · · · ⊃ S5 . Definition 2.9. Let ( W, R ) be an S5 -frame. Then it is a disjoint union of equiva-lence classes, or clusters , S ι<κ C ι , where each C ι is of the form R ( { w } ) (which wehenceforth write as R ( w )) for some w ∈ W . We define: • the number of clusters to be κ ; • the lower cluster size to be min ι<κ | C ι | , and • the upper cluster size to be sup ι<κ | C ι | .The unique (up to isomorphism) frame with one cluster of size λ is the λ -cluster, and we denote it by C λ .Theorem 2.7 can be sharpened using the structures C λ . Theorem 2.10. For each finite n > , S5 n is sound and strongly complete for the n -cluster, while for infinite λ , S5 λ is sound and strongly complete for any κ -clusterwhere κ ≥ λ .Proof sketch. It follows from a result of Scroggs that S5 n is sound and complete forthe n -cluster. The proof of strong completeness is along the same lines as that ofTheorem 2.7. For infinite λ , we have that S5 λ is sound and strongly complete forthe class of S5 -frames. So, if Γ ϕ , there is a model M and a world w of M so that M | = Γ but w J ϕ K . Then, the generated submodel of w is its cluster and satisfiesthe same formulas. Using the upwards or downwards L¨owenheim-Skolem theorem,we can assume that this cluster is isomorphic to C κ . (cid:3) Baire algebras as a new semantics for S5 In this section we introduce the main concept of the paper, that of Baire algebras,which provide a new semantics for S5 . Baire algebras are obtained from topologicalspaces X by modding out ℘ ( X ) by the σ -ideal M of meager subsets of X . We showthat the closure operator c on X gives rise to a closure operator on ℘ ( X ) / M , sothat ( ℘ ( X ) / M , c ) is a monadic algebra.We start by the following well-known definition. Definition 3.1. Let X be a topological space and A ⊆ X .(1) A is nowhere dense if ic A = ∅ .(2) A is meager if it is a σ -union of nowhere dense sets.(3) M denotes the set of meager subsets of X .(4) X is a Baire space if for each nonempty open subset U of X , we have that U / ∈ M .It is easy to see that M is a σ -ideal of ℘ ( X ). Therefore, the quotient ℘ ( X ) / M is a σ -algebra (see, e.g., [16, Sec. 13]). However, ℘ ( X ) / M is not always a completeBoolean algebra (see, e.g., [16, Sec. 25]).We recall that elements of ℘ ( X ) / M are equivalence classes of the equivalencerelation given by A ≈ B iff A \ B, B \ A ∈ M . G. BEZHANISHVILI AND D. FERNANDEZ-DUQUE We write [ A ] for the equivalence class of A under ≈ . Then [ A ] ⊑ [ B ] iff A \ B ∈ M ,and the operations on ℘ ( X ) / M are given by(1) l n<ω [ A n ] = [ \ n<ω A n ] and − [ A ] = [ X \ A ] . On the other hand, as we already pointed out above, if S ⊆ ℘ ( X ) is uncountable,then d S may not always exist. Example 3.2. Let ( W, R ) be an S4 -frame, which as usual we identify with thetopological space whose opens are those U ⊆ W that satisfy R ( U ) = U . A point w ∈ W is quasimaximal if wRv implies vRw . Let qmax W be the set of quasimaximalpoints. We have that w ∈ qmax W iff R ( w ) ⊆ R − ( w ). Since R ( w ) is the leastopen containing w , we obtain that w ∈ qmax W iff { w } is not nowhere dense.Thus, if W is countable, then W \ qmax W is the largest meager set, and hence ℘ ( W ) / M is isomorphic to ℘ ( qmax W ). In particular, if ( W, R ) is an S5 -frame, then W = qmax W , and so ℘ ( W ) / M is isomorphic to ℘ ( W ). Definition 3.3. Let X be a topological space.(1) We call a ∈ ℘ ( X ) / M open if a = [ U ] for some open U ∈ ℘ ( X ), and closed if a = [ F ] for some closed F ∈ ℘ ( X ). An element that is both open andclosed is clopen. (2) We denote the set of closed elements of ℘ ( X ) / M by Γ.It is easy to see that Γ is a bounded sublattice of ℘ ( X ) / M . We next show thatΓ is in fact a Boolean subalgebra of ℘ ( X ) / M . For this we point out that if U isopen, then c U \ U is nowhere dense, so [ c U ] = [ U ]; and similarly, if F is closed,then [ F ] = [ i F ]. Theorem 3.4. For every topological space X , we have that Γ is a Boolean subal-gebra of ℘ ( X ) / M .Proof. Since Γ is a bounded sublattice of ℘ ( X ) / M , it is sufficient to show that a ∈ Γ implies − a ∈ Γ. From a ∈ Γ it follows that a = [ F ] for some F closed in X .Then − a = [ X \ F ]. Since X \ F is open, we have that X \ F ≈ c ( X \ F ). Thus, − a = [ X \ F ] = [ c ( X \ F )], and hence − a ∈ Γ. (cid:3) We next wish to prove that Γ is a relatively complete subalgebra of ℘ ( X ) / M .As we pointed out in the Introduction, this will follow from the Banach CategoryTheorem (see, e.g., [24, Thm. 16.1]), which we recall next. Theorem 3.5 (Banach Category Theorem) . If X is a topological space and O isa collection of meager open subsets of X , then S O is meager. The following can then be seen as a dual version of Theorem 3.5. Theorem 3.6. For an arbitrary topological space X , we have that Γ is a relativelycomplete subalgebra of ℘ ( X ) / M . Before proving the theorem, we recall some preliminary notions. Let A ⊆ X . Aset U ⊆ X is relatively open in A if U ∩ A is open in A , seen as a subspace of X . Thenotions of relatively closed and relatively meager are defined analogously. Obviously, B is relatively open in A iff there is an open set U such that U ∩ A = B ∩ A , andrelatively closed iff there is closed C such that C ∩ A = B ∩ A . HE BAIRE CLOSURE AND ITS LOGIC 9 Lemma 3.7. Let X be a topological space and A, B ⊆ X . If B is relatively meagerin A , then B ∩ A is meager in X . Before giving the proof, we remark that the converse is not true in general (forexample, we may have B ∩ A = ∅ even if B is non-meager). Proof. By definition, if B is relatively meager in A then there are relatively closedsets M n such that B ∩ A ⊆ S n M n and whenever U is open and U ∩ A = ∅ , then U ∩ A M n . Note that we may assume that the sets M n are already closed in X ,otherwise replace M n by a closed set M ′ n with M ′ n ∩ A = M n ∩ A . We may furtherassume that M n ⊆ c A , as otherwise we may replace M n by M n ∩ c A . Under theseassumptions, we claim that each M n is nowhere dense. Let U ⊆ X be open. If U ∩ c A = ∅ , then U ∩ M n = ∅ , so we may assume that U ∩ c A = ∅ . But then, U ∩ A = ∅ , and by assumption U ∩ A M n , so U M n . Since U was arbitrary, M n is nowhere dense. Hence we conclude that if B ⊆ X is relatively meager in A ,it follows that there are nowhere dense sets M n such that B ∩ A ⊆ S n M n , i.e., B ∩ A is meager. (cid:3) Proof of Theorem 3.6. Let a = [ A ] ∈ ℘ ( X ) / M and Γ a be the collection of all closedsets C such that [ A ] ⊑ [ C ]. We claim that T Γ ∈ Γ. Let ∆ a = { X \ C : C ∈ Γ a } .Each U ∈ ∆ a is open (hence relatively open) and relatively meager. It follows fromTheorem 3.5 that U ∗ = S ∆ a is relatively meager in A . By Lemma 3.7, U ∗ ∩ A is meager. Letting C ∗ := T Γ a we see that A \ C ∗ = U ∗ ∩ A , and since the latteris meager, we conclude that C ∗ ∈ Γ a . Since clearly [ C ∗ ] is then the least closedelement containing a , the theorem follows. (cid:3) Remark 3.8. If X is a Baire space, then Γ is even a complete Boolean algebra.Indeed, Γ is isomorphic to the quotient of all Borel subsets of X by M , which inturn is isomorphic to the algebra of all regular open subsets of X ; see, e.g., [18,Prop. 12.9].As an immediate consequence of Theorems 3.4 and 3.6, we obtain: Theorem 3.9. If X is a topological space, then Baire ( X ) := ( ℘ ( X ) / M , c ) is amonadic algebra, where for a ∈ ℘ ( X ) / M , c a is defined as the least closed elementabove a . Remark 3.10. In general, for a closure algebra ( B, c ), an element a ∈ B is closed if a = c a , open if a = i a , and clopen if it is both closed and open. Since Baire ( X )is a closure algebra, it is not hard to check that these notions as given in Definition3.3 coincide with the general algebraic definition.4. Baire resolvability To obtain completeness results with respect to this new semantics we require tointroduce the concept of Baire resolvability, which refines the concept of resolvabilityintroduced by Hewitt [17]. A space is resolvable if it can be partitioned into disjointdense sets. This notion readily extends to the algebraic setting: Definition 4.1. Let B be a Boolean algebra. We call a, b ∈ B orthogonal if a ∧ b = 0, and a family { a i | i ∈ I } pairwise orthogonal if a i ∧ a j = 0 for i = j . Definition 4.2. Let ( B, c ) be a closure algebra and κ a nonzero cardinal.(1) We call a ∈ B dense if c a = 1. (2) We call ( B, c ) κ -resolvable if there is a pairwise orthogonal family of denseelements whose cardinality is κ . Definition 4.3. Let ( B, c ) be a closure algebra, a ∈ B , and κ a nonzero cardinal.We call a κ -resolvable if there is a pairwise orthogonal family { b ι | ι < κ } such that b ι ≤ a and a ≤ c b ι for each ι < κ .If { [ A ι ] | ι < κ } is a κ -resolution of Baire ( X ), we remark that the collectionof representatives A ι need not come from a partition of X ; for example, two ofthem may have nonempty (but meager) intersection. However, sometimes it willbe useful to have ‘nice’ representatives, as made precise in the following definition. Definition 4.4. Let X be a Baire space. For a nonzero cardinal κ , we call Y ⊆ X Baire κ -resolvable if there is a partition { A ι | ι < κ } of Y such that, for each ι , c [ A ι ] = [ Y ]. We call { A ι | ι < κ } a Baire κ -resolution of Y . If { A ι | ι < κ } is a Baire κ -resolution of Y , then { [ A ι ] | ι < κ } is a κ -resolutionof [ Y ]. Thus, the Baire κ -resolvability of Y implies the κ -resolvability of [ Y ]. Aswe have mentioned, the converse is not true in principle since the A ι do not have toform a partition, but it is not hard to check that the two notions are equivalent if κ is countable. Note that a ι = [ A ι ] is dense in Baire ( X ) provided for each nonemptyopen U ∈ ℘ ( X ) we have A ι ∩ U / ∈ M . We call such sets ‘nowhere meager’. Moregenerally, we adopt the following conventions. Definition 4.5. Let X be a topological space and κ a cardinal. We say that: • A ⊆ X is nowhere meager if for each nonempty open U ⊆ X we have that A ∩ U / ∈ M . • A ⊆ X is somewhere meager if it is not nowhere meager. • X is everywhere Baire κ -resolvable if each U ⊆ X is Baire κ -resolvable (seenas a subspace of X ). • X is somewhere Baire κ -resolvable if there is some U ⊆ X that is Baire κ -resolvable. • X is nowhere Baire κ -resolvable if it is not somewhere Baire κ -resolvable.So, A ⊆ X is somewhere meager iff there is a nonempty open U ⊆ X with A ∩ U ∈ M . Clearly A is nowhere meager iff for each proper closed F ∈ ℘ ( X ) wehave A \ F / ∈ M . This yields the following characterizations of Baire resolvability: Lemma 4.6. A Baire space X is Baire κ -resolvable if and only if it can be parti-tioned into κ many nowhere meager sets. Lemma 4.7. Let X be a topological space and κ a nonzero cardinal. Then X isBaire κ -resolvable iff it is everywhere Baire κ -resolvable.Proof. Clearly if X is everywhere Baire κ -resolvable, then X is Baire κ -resolvable.Conversely, suppose that X is Baire κ -resolvable and { A ι | ι < κ } is a Baire κ -resolution of X . Let U ⊆ X be nonempty open. Since A ι is nowhere meager, A ι ∩ U is nowhere meager in U . Thus, { A ι ∩ U | ι < κ } is a Baire κ -resolution of U . (cid:3) Let X be a Baire space and κ ≤ ω . It is not hard to see that if X is Baire κ -resolvable, then X is κ -resolvable. The next example shows that the converse isnot true in general. HE BAIRE CLOSURE AND ITS LOGIC 11 Example 4.8. Let X = ω + 1 with the topology whose nonempty open sets areof the form [ n, ω ] where n < ω . Then { ω } is dense in X , and being a singleton,it cannot be written as a countable union of nowhere dense sets, so { ω } / ∈ M .Therefore, every nonempty open set in X is not meager, and hence X is a Bairespace.Each infinite subset of [0 , ω ) is dense. Therefore, if we write [0 , ω ) as a countableunion of infinite disjoint subsets of [0 , ω ), we obtain that X is ω -resolvable. On theother hand, [0 , ω ) = S n<ω [0 , n ] and each [0 , n ] is nowhere dense, so [0 , ω ) ∈ M .Thus, no subset of [0 , ω ) is nowhere meager, yielding that X is not Baire resolvable.As we pointed out in the Introduction, our main Baire resolvability results followfrom the Disjoint Refinement Lemma (see [9, Lem. 7.5] and [9, Notes for § 7] for thehistory of this result). This lemma has been used in proofs of resolvability in othercontexts (see e.g. [8]); ours follows a similar pattern. Lemma 4.9 (Disjoint Refinement Lemma) . Let κ be an infinite cardinal and { A ι | ι < κ } a family of sets such that | A ι | = κ for each ι < κ . Then there is a family { B ι | ι < κ } of pairwise disjoint sets such that B ι ⊆ A ι and | B ι | = κ for each ι < κ . We will use the following version of the Disjoint Refinement Lemma: Lemma 4.10. Let κ be an infinite cardinal, X a set, and K a family of subsetsof X such that |K| ≤ κ and | K | ≥ κ for each K ∈ K . Then there is a partition { A ι | ι < κ } of X such that A ι ∩ K = ∅ for each K ∈ K and ι < κ .Proof. Enumerate K as { K ι | ι < κ } , and for each ι < κ let K ′ ι be a subset of K ι with | K ′ ι | = κ . By the Disjoint Refinement Lemma, there is a family { B ι | ι < κ } ofpairwise disjoint sets such that B ι ⊆ K ′ ι and | B ι | = κ for each ι < κ . Enumeratingeach B ι as { b ιν | ν < κ } , for ν < κ , let A ′ ν = { b ιν | ι < κ } . To ensure that weobtain a partition, let A = A ′ ∪ (cid:0) X \ S ν<κ A ′ ν (cid:1) and for ν > 0, let A ν = A ′ ν .Then { A ν | ν < κ } is the desired partition, since for each ι < κ we have that b ιν ∈ A ν ∩ K ′ ι ⊆ A ν ∩ K ι . (cid:3) To apply Lemma 4.10 to Baire resolvability, we use the notion of a κ -witnessingfamily. Definition 4.11. Let X be a Baire space and κ a cardinal. We call K ⊆ ℘ ( X ) a κ -witnessing family for X provided |K| ≤ κ , | K | ≥ κ for each K ∈ K , and if A ⊆ X is somewhere meager, then there is K ∈ K with K ∩ A = ∅ . Theorem 4.12. Let κ be an infinite cardinal. Each Baire space that has a κ -witnessing family is Baire κ -resolvable.Proof. Let X be a Baire space with a κ -witnessing family K . By Lemma 4.10, thereis a family { A ι | ι < κ } of pairwise disjoint subsets of X such that A ι ∩ K = ∅ foreach K ∈ K and ι < κ . We claim that each A ι is nowhere meager. Otherwise, since K is a κ -witnessing family, there is K ∈ K such that K ∩ A ι = ∅ , a contradiction.Thus, { A ι | ι < κ } gives the desired Baire κ -resolution of X . (cid:3) In the next two lemmas we show that the uncountable compact sets provide awitnessing family in many Baire spaces. These lemmas are folklore, but we statethem in the precise form we need and briefly recall their proofs. Lemma 4.13. Let κ = 2 ℵ and X be a crowded topological space which is eitherlocally compact Hausdorff or completely metrizable. If A ⊆ X is somewhere meager,then there is a compact K ⊆ X such that | K | ≥ c and A ∩ K = ∅ .Proof. The proof follows the standard tree construction method (see, e.g., [10,Exercise 3.12.11]). We provide a sketch in the case when X is a crowded locallycompact Hausdorff space. Suppose that A ⊆ X is somewhere meager, so there isa nonempty open U ⊆ X such that A ∩ U ∈ M . Then A ∩ U = S n<ω B n , witheach B n nowhere dense. Using the assumption that X is crowded locally compactHausdorff, to each finite sequence b = ( b , . . . , b n ) ∈ { , } <ω , we can assign acompact set K b ⊆ U so that • K b has nonempty interior; • B i ∩ K b = ∅ for each i ≤ n ; • if b is an initial segment of b ′ , then K b ′ ⊆ K b ; • if b , b ′ are incomparable (i.e. disagree on some coordinate), then K b ∩ K b ′ = ∅ .Let K = T n<ω S | b | = n K b . Being an intersection of compact sets, this is a com-pact set. It follows from the construction that each K b is contained in U and that A ∩ K = ∅ . The intersections along infinite sequences b ∈ { , } ω are nonemptyand disjoint, witnessing that | K | ≥ c .The proof for a crowded completely metrizable X is essentially the same, but weneed to ensure that the diameter of each K b is at most 2 −| b | so that we can usethat X is complete. (cid:3) Lemma 4.14. Let X be be either Hausdorff and second-countable or metrizableand of cardinality continuum. Then X contains at most c compact sets.Proof. First suppose that X is Hausdorff and second-countable. The latter impliesthat X contains at most c open sets, and so at most c closed sets. Since X isHausdorff, compact sets are closed, and hence there can be at most c compact sets.Next suppose that X is metrizable and of cardinality continuum. Then everycompact set in K ⊆ X is closed and totally bounded, from which it readily followsthat K is the closure of some countable subset of X . Since X has c -many countablesubsets, there can be at most c values for K . (cid:3) Putting together these results, we arrive at the following: Theorem 4.15. Suppose that X is a crowded space which is either locally compact,Hausdorff, and second-countable or completely metrizable and continuum sized.Then X is Baire c -resolvable.Proof. By Lemmas 4.13 and 4.14, the uncountable compact subsets of X form a c -witnessing family for X . Therefore, by Theorem 4.12, X is Baire c -resolvable. (cid:3) Remark 4.16. Lemma 4.13 can be generalized to the setting of ˇCech-completespaces, under the assumption that no point has a least neighborhood (equivalently,that every point has infinite character: see, e.g., [10, Exercise 3.12.11(b)]). Thisallows us to extend Theorem 4.15 to include any second-countable ˇCech-completespace with the latter property. HE BAIRE CLOSURE AND ITS LOGIC 13 New completeness results for S5 In this section we derive new completeness results for S5 and its extensions usingour new semantics of Baire algebras. We start by recalling that a map h : A → B between two closure algebras ( A, c A ) and ( B, c B ) is a homomorphism of closurealgebras if h is a homomorphism of Boolean algebras and h ( c A a ) = c B h ( a ) for each a ∈ A . We say that h is an embedding if it is injective, and that h is an isomorphism if it is bijective.Recall that if X and Y are topological spaces and f : X → Y , then f is continuous if f − ( B ) is open whenever B ⊆ Y is open, open if f ( A ) is open whenever A ⊆ X is open, and an interior map if it is both continuous and open. It is well known(see, e.g., [25, p. 99]) that if f : X → Y is an interior map, then f − : Kur ( Y ) → Kur ( X ) is a homomorphism of closure algebras. Moreover, if f is surjective, then f − is an embedding. To obtain analogous results for Baire algebras, it is sufficientto work with partial maps. Below, recall that [ A ] denotes the equivalence class of A modulo the meager sets. Definition 5.1. Let X, Y be arbitrary topological spaces and f : X → Y a partialmap.(1) Say that f is defined almost everywhere if the complement of the domainof f is meager.(2) Call f proper if B ⊆ Y meager implies that f − ( B ) is meager.(3) Define h : Baire ( Y ) → Baire ( X ) by h [ A ] = [ f − ( A )] for each A ⊆ Y. Lemma 5.2. Let X, Y be arbitrary topological spaces and f : X → Y a partial map.If f is defined almost everywhere and is proper, then h : Baire ( Y ) → Baire ( X ) isa well-defined homomorphism of Boolean algebras.Proof. Let A, B ⊆ Y with [ A ] = [ B ]. Then A \ B is meager, so f − ( A ) \ f − ( B ) = f − ( A \ B ) is meager since f is proper. By a symmetric argument, f − ( B ) \ f − ( A )is also meager. Thus, [ f − ( A )] = [ f − ( B )], and hence h is well defined.In addition, h ([ A ] ⊓ [ B ]) = [ f − ( A ∩ B )] = [ f − ( A )] ⊓ [ f − ( B )] = h ( A ) ⊓ h ( B ) . Therefore, h commutes with binary meets. To see that it also commutes withcomplements, let M be the complement of the domain of f . Then X \ f − ( A ) = f − ( Y \ A ) ∪ M . Since f is defined almost everywhere, M is meager, so [ X \ f − ( A )] = [ f − ( Y \ A )]. Thus, − h [ A ] = [ X \ f − ( A )] = [ f − ( Y \ A )] = h ( − [ A ]) . Consequently, h is a well-defined homomorphism of Boolean algebras. (cid:3) Note, however, that h may fail to be injective. To obtain an embedding ofBoolean algebras, we need an additional condition on f . Definition 5.3. We call a partial map f : X → Y exact if f − ( A ) meager impliesthat A is meager for each A ⊆ Y . Lemma 5.4. Let X, Y be Baire spaces and let f : X → Y be proper, exact, anddefined almost everywhere. Then h : Baire ( Y ) → Baire ( X ) is an embedding ofBoolean algebras. Proof. By Lemma 5.2, h is a homomorphism of Boolean algebras. To see that it isan embedding, let h [ A ] = h [ B ]. Then f − ( A ) \ f − ( B ) = f − ( A \ B ) is meager.Since f is exact, A \ B is meager. A symmetric argument yields that B \ A ismeager. Thus, [ A ] = [ B ], and hence h is an embedding. (cid:3) Next we detail some conditions that will ensure that h is not only a Booleanhomomorphism, but indeed a homomorphism of closure algebras. Definition 5.5. Let f : X → Y be a partial map. We say that(1) f is Baire-continuous if V an open subset of Y implies that [ f − ( V )] is anopen element of Baire ( X );(2) f is Baire-open if U a nonempty open subset and M a meager subset of X imply that [ f ( U \ M )] is a nonzero open element of Baire ( Y );(3) f is a Baire map if it is defined almost everywhere and is proper, Baire-continuous, and Baire-open. Remark 5.6. If a Baire map f : X → Y exists, then X must be a Baire space.This is because if U ⊆ X were a nonempty open meager set, then [ f ( U \ U )] = 0in Baire ( Y ), so f could not be Baire-open. Lemma 5.7. Let X, Y be topological spaces. If f : X → Y is a Baire map, then h : Baire ( Y ) → Baire ( X ) is a homomorphism of closure algebras. If in addition f is exact, then h is an embedding of closure algebras.Proof. By Lemma 5.2, h is a homomorphism of Boolean algebras. Let A ⊆ Y . Wefirst show that c h [ A ] ⊑ h c [ A ]. There is a closed set D ⊆ Y such that [ D ] = c [ A ].Since h is a homomorphism of Boolean algebras, it is order preserving. Therefore,[ A ] ⊑ c [ A ] implies h [ A ] ⊑ h c [ A ] = [ f − ( D )]. Let U = Y \ D and recall that M isthe complement of the domain of f . Since U is open and M is meager in Y , wehave[ f − ( D )] = [ f − ( Y \ U )] = [ f − ( Y \ U ) ∪ M ] = [ X \ f − ( U )] = − [ f − ( U )] . Because f is Baire-continuous and U is open, [ f − ( U )] is open, hence [ f − ( D )] isclosed in Baire ( X ). Thus, c h [ A ] ⊑ [ f − ( D )] = h c [ A ] by definition of closure.It is left to show that h c [ A ] ⊑ c h [ A ]. There is a closed set C ⊆ X such that c h [ A ] = [ C ]. Let D be as above and let V = i f − ( D ) \ C . Since f − ( D ) is closedin X , we have [ i f − ( D )] = [ f − ( D )]. Therefore,[ V ] = [ f − ( D )] − [ C ] = h c [ A ] − c h [ A ] , so it suffices to show that V = ∅ . Suppose otherwise. Since [ f − ( A )] ⊑ [ C ], wehave that f − ( A ) \ C is meager, so f − ( A ) ∩ V is meager. Because f is Baire-open and V is nonempty open, [ f ( V ) \ A ] = [ f ( V \ f − ( A ))] is nonzero and openin Baire ( Y ). Thus, c [ A ] − [ f ( V ) \ A ] ⊏ c [ A ]. Recall that we chose D so that c [ A ] = [ D ]. In particular, [ A ] ⊑ [ D ]. Since A \ ( D \ ( f ( V ) \ A )) = A \ D and thelatter is meager, we see that[ A ] ⊑ [ D \ ( f ( V ) \ A )] = c [ A ] − [ f ( V ) \ A ] ⊏ c [ A ] . As c [ A ] − [ f ( V ) \ A ] is closed in Baire ( Y ), this contradicts the definition of c [ A ].Consequently, V = ∅ , as required.Finally, if f is exact, then it follows from Lemma 5.4 that h is an embedding. (cid:3) HE BAIRE CLOSURE AND ITS LOGIC 15 If Y is an S5 -frame, it follows from Example 3.2 that Kur ( Y ) is isomorphic to Baire ( Y ), so we may regard h as a map from Kur ( Y ) to Baire ( X ). Since theonly meager subset of an S5 -frame is ∅ , the next lemma is immediate. Lemma 5.8. Let X be a Baire space, Y an S5 -frame, and f : X → Y a Baire map.Then f is exact iff f − ( y ) is non-meager for each y ∈ Y . Recall that for a nonzero cardinal κ , C κ denotes the κ -cluster. By [2, Lem. 5.9],a space X is κ -resolvable iff there is an interior map from X onto C κ . We have thefollowing analogue of this result for Baire algebras. Lemma 5.9. Let X be a Baire space and κ a nonzero cardinal. Then X is Baire κ -resolvable iff there is an exact Baire map f : X → C κ .Proof. Let ( w ι ) ι<κ be an enumeration of C κ . First suppose that X is Baire κ -resolvable, and let ( A ι ) ι<κ be a Baire κ -resolution of X . Define f : X → C κ bysetting f ( x ) = w ι provided x ∈ A ι . Then f is a total onto map. Since f is total,it is defined almost everywhere, and f is proper because ∅ is the only meagersubset of C κ . Since ∅ , C κ are the only opens of C κ , it is clear that f is continuous,hence Baire-continuous. To see that f is Baire-open, let U ⊆ X be nonemptyopen and M ⊆ X meager. Each A ι ∩ ( U \ M ) is non-meager, hence nonempty,so w ι ∈ f ( U \ M ), and so f ( U \ M ) = C κ . Therefore, f is a Baire map. Finally, f − ( w ι ) = A ι , which is non-meager. Thus, f is exact by Lemma 5.8.Next suppose that f : X → C κ is an exact Baire map. By Lemma 5.8, each f − ( w ι ) is non-meager, and we can enlarge f − ( w ) to obtain a partition of X if needed. By identifying Baire ( C κ ) with Kur ( C κ ) and applying Lemma 5.7, h : Kur ( C κ ) → Baire ( X ) is a homomorphism of closure algebras. Therefore, for each ι < κ , c [ f − ( w ι )] = c h ( w ι ) = h c ( w ι ) = h ( C κ ) = [ f − ( C κ )] = [ X ] . Thus, ( f − ( w ι )) ι<κ witnesses that X is Baire κ -resolvable. (cid:3) We are ready to prove our first completeness result. For this we recall that S5 λ denotes the variant of S5 with λ -many variables, and if λ < ω , then S5 λ is theextension of S5 axiomatized by Alt λ . Theorem 5.10. Let X be a Baire space, λ a nonzero cardinal, and suppose that X is Baire λ -resolvable. (1) If λ ≥ ω , then S5 λ is sound and strongly complete for Baire ( X ) . (2) If λ < ω and X is nowhere Baire ( λ + 1) -resolvable, then S5 λ is sound andstrongly complete for Baire ( X ) .Proof. (1) Soundness follows from the fact that Baire ( X ) is a monadic algebra (see The-orem 3.9). For completeness, let Γ | = Baire ( X ) ϕ . By Lemma 5.9, there is an exactBaire map f : X → C λ . Since Kur ( C λ ) is isomorphic to Baire ( C λ ), by Lemma 5.7, Kur ( C λ ) embeds into Baire ( X ). Therefore, Γ | = C λ ϕ . Thus, Γ ⊢ S5 λ ϕ by Theo-rem 2.7(2).(2) We first prove completeness. Suppose Γ | = Baire ( X ) ϕ . By Lemma 5.9, there isan exact Baire map f : X → C λ . By Lemma 5.7, Kur ( C λ ) embeds into Baire ( X ).Therefore, Γ | = C λ ϕ . Thus, Γ ⊢ S5 λ ϕ by Theorem 2.10. To prove soundness, suppose that S5 λ is not sound for X , and fix a valuation J · K falsifying the S5 λ axiom Alt λ . Since this axiom is a Boolean combination of formulas ♦ ψ , we have that J Alt λ K is clopen, so we can choose U open with [ U ] = J ¬ Alt λ K .Then( ∗ ) [ U ] ⊑ J λ ^ i =0 ♦ p i K and( † ) [ U ] ⊓ J _ ≤ i Corollary 5.11. Let X be a crowded space which is either completely metrizableand of cardinality continuum, or second-countable locally compact Hausdorff. Then S5 c is sound and strongly complete for Baire ( X ) . In particular, we obtain: Corollary 5.12. S5 is the logic of Baire ( X ) whenever X is: (1) R n for any n ≥ . (2) Any perfect closed subset of R n , including the Cantor space and the interval [0 , . (3) The Banach space ℓ p for p ∈ [1 , ∞ ] . Note that in the third item we may allow p = ∞ since | ℓ ∞ | = c , despite notbeing second-countable. Remark 5.13. On the other hand, since Q is meager, Baire ( Q ) is the trivialalgebra, and hence its logic is the contradictory logic, i.e. the logic axiomatized by ⊥ . This is in contrast with the McKinsey-Tarski theorem that the logic of Kur ( Q )is S4 .By Theorem 5.10(2), if X is a Baire space that is Baire n -resolvable and nowhereBaire ( n + 1)-resolvable, then the logic of Baire ( X ) is S5 n . As we next show, wecan avoid verifying whether X is nowhere Baire ( n + 1)-resolvable by constructinga subalgebra of Baire ( X ) whose logic is S5 n . For this we require the followingdecomposition lemma, which follows from similar decompositions in the theory of ℓ -groups; see, e.g., [5] and the references therein. To keep the paper self-contained,we give a proof. Lemma 5.14. Let A be a monadic algebra, A its subalgebra of all clopen elements, B a Boolean subalgebra of A , and C the Boolean subalgebra of A generated by A ∪ B . (1) C is a monadic subalgebra of A . HE BAIRE CLOSURE AND ITS LOGIC 17 (2) Each c ∈ C can be written as c = W ni =1 ( a i ∧ b i ) , where a , . . . , a n ∈ A arepairwise orthogonal and b , . . . , b n ∈ B . (3) Each c , . . . , c n ∈ C can be written in the compatible form c j = W ni =1 ( a i ∧ b ji ) where a , . . . , a n ∈ A are pairwise orthogonal and each b ji ∈ B .Proof. (1) For a ∈ C we have c a ∈ A ⊆ C . Thus, C is a monadic subalgebra of A .(2) Since C is a Boolean subalgebra of A , it is well known (see, e.g., [25, p. 74])that each c ∈ C can be written as c = W ni =1 V m j j =1 d ij where either d ij ∈ A ∪ B or − d ij ∈ A ∪ B . Since both A and B are Boolean subalgebras, we may assumethat d ij ∈ A ∪ B ; and gathering together the elements in A and B , we may write V m j j =1 d ij = a i ∧ b i where a i ∈ A and b i ∈ B . Therefore, each c ∈ C can be writtenas c = W ni =1 ( a i ∧ b i ) where a i ∈ A and b i ∈ B . It is left to prove that the a i can be chosen pairwise orthogonal. But this is a standard argument. Indeed, if c = ( a ∧ b ) ∨ ( a ∧ b ), then we may write c = ( a ∧ b ) ∨ ( a ∧ b )= [(( a − a ) ∧ b ) ∨ (( a ∧ a ) ∧ b )] ∨ [(( a ∧ a ) ∧ b ) ∨ (( a − a ) ∧ b )]= (( a − a ) ∧ b ) ∨ (( a ∧ a ) ∧ ( b ∨ b )) ∨ (( a − a ) ∧ b ) , where a − a , a ∧ a , a − a ∈ A are pairwise orthogonal and b , b ∨ b , b ∈ B .Now a simple inductive argument finishes the proof.(3) We only consider the case of two elements c, d as the general case follows bysimple induction. By (2), we may write c = W ni =1 ( a i ∧ b i ) where a , . . . , a n ∈ A are pairwise orthogonal and b , . . . , b n ∈ B . Similarly d = W mj =1 ( e j ∧ f j ) where e , . . . , e m ∈ A are pairwise orthogonal and f , . . . , f m ∈ B . By letting a n +1 = ¬ W ni =1 a i and b n +1 = 0 we may assume that W ni =1 a i = 1, and similarly W mj =1 e j = 1.But then c = n _ i =1 m _ j =1 (( a i ∧ e j ) ∧ b i ) and d = m _ j =1 n _ i =1 (( a i ∧ e j ) ∧ f j ) . Clearly a ∧ e , . . . , a ∧ e m , . . . , a n ∧ e , . . . , a n ∧ e m are pairwise orthogonal and are in A ; also each of b i , f j are in B (and may appearmultiple times in the decomposition). (cid:3) Theorem 5.15. Let X be a Baire space and n < ω . If X is Baire n -resolvable,then Baire ( X ) has a subalgebra A containing all clopens of Baire ( X ) such that S5 n is sound and strongly complete for A .Proof. Let C n = { w , . . . , w n } be the n -cluster. By Lemmas 5.7 and 5.9, there isan embedding h : Kur ( C n ) → Baire ( X ). Let A = h [ Kur ( C n )] and H = { h ( w ) | w ∈ C n } . Then A is a monadic subalgebra of Baire ( X ) generated by H , and each h ( w ) is an atom of A . Let B be the Boolean subalgebra of Baire ( X ) generatedby A and the clopen elements of Baire ( X ). By Lemma 5.14(1), B is a monadicsubalgebra of Baire ( X ). We claim that the logic of B is S5 n .Since S5 n is complete for C n , it follows that S5 n is complete for B . It remainsto check that S5 n is also sound for B . For this it is sufficient to show that Alt n isvalid on B . Let J · K be a valuation on B . By Lemma 5.14(3), we may write each J p i K in the form F mℓ =1 ( u ℓ ⊓ a iℓ ), where the u ℓ are orthogonal clopens and the a iℓ areelements of A . Therefore, J n +1 ^ i =1 ♦ p i K = n +1 l i =1 m G ℓ =1 c ( u ℓ ⊓ a iℓ ) = n +1 l i =1 m G ℓ =0 ( u ℓ ⊓ c a iℓ ) = m G ℓ =1 ( u ℓ ⊓ n +1 l i =1 c a iℓ ) , where the last equality uses distributivity plus the orthogonality of the u ℓ , so thatall cross-terms cancel. Similarly, J _ i = j ♦ ( p i ∧ p j ) K = G i = j m G ℓ =1 ( u ℓ ⊓ c ( a iℓ ⊓ a jℓ )) = m G ℓ =1 ( u ℓ ⊓ G i = j c ( a iℓ ⊓ a jℓ )) . So to show that J ϕ K = 1, it suffices to show that(2) m G ℓ =1 ( u ℓ ⊓ n +1 l i =1 c a iℓ ) ⊑ m G ℓ =1 ( u ℓ ⊓ G i = j c ( a iℓ ⊓ a jℓ )) . Given that the u ℓ are orthogonal, (2) holds iff, for each ℓ , u ℓ ⊓ n +1 l i =1 c a iℓ ⊑ u ℓ ⊓ G i = j c ( a iℓ ⊓ a jℓ ) , and for this it suffices to show that(3) n +1 l i =1 c a iℓ ⊑ G i = j c ( a iℓ ⊓ a jℓ ) . So fix ℓ . Note that (3) holds trivially if a iℓ = 0 for some i (as the left-hand side isthen zero), so we assume otherwise. Since A is generated by H , for each i there is w ∈ C n such that h ( w ) ⊑ a iℓ . Because | C n | = n , the pigeonhole principle yields thatthere are i = j and w so that h ( w ) ⊑ a iℓ ⊓ a jℓ . But c h ( w ) = 1, so c ( a iℓ ⊓ a jℓ ) = 1.Therefore, n +1 l i =1 c a iℓ ⊑ c ( a iℓ ⊓ a jℓ ) ⊑ G i = j c ( a iℓ ⊓ a jℓ ) . Thus, (3) holds and, since ℓ was arbitrary, we conclude that (2) holds, as needed. (cid:3) As a consequence of Theorems 5.15 and 4.15 we obtain: Corollary 5.16. Let X be a crowded space which is either a complete metricspace of cardinality c or second-countable locally compact Hausdorff. Then for eachnonzero n < ω there is a subalgebra of Baire ( X ) containing all clopen elementssuch that S5 n is sound and strongly complete for Baire ( X ) . New completeness results for S5U In this section, we extend our completeness results to the language with theuniversal modality. As in the previous section, these results utilize standard Kripkecompleteness results. In particular, the following completeness result is well known(see, e.g., [14]), and can be lifted to strong completeness via standard methods. Theorem 6.1. S5U is sound and strongly complete for the class of countable S5 -frames. HE BAIRE CLOSURE AND ITS LOGIC 19 In the topological setting, the language L ♦ ∀ is able to express connectedness byShehtman’s axiom ∀ ( (cid:3) p ∨ (cid:3) ¬ p ) → ∀ p ∨ ∀¬ p (see [26]). Therefore, S4U is not complete for any connected space. The situationis quite different in the setting of Baire algebras, and in order to see this, we willneed to discuss disconnectedness in this context. This notion has already beenconsidered in the context of closure algebras by McKinsey and Tarski [23]. Definition 6.2. A closure algebra ( B, c ) is disconnected if there are open a, b ∈ B such that a, b = 0, a ∧ b = 0, and a ∨ b = 1. Lemma 6.3. If X is a Hausdorff Baire space with at least two points, then Baire ( X ) is disconnected.Proof. Let x = y in X , and let U, V be the disjoint neighborhoods of x, y . Set a = [ U ]. Since U is open, a is open, hence clopen in Baire ( X ). Because X isa Baire space, U is non-meager, so a = 0. Also, X \ U is non-meager because V ⊆ X \ U . Thus, a = 1, and hence Baire ( X ) is disconnected. (cid:3) As was the case with resolvability, disconnectedness readily extends to κ -dis-connectedness for any cardinal κ . Definition 6.4. A closure algebra ( B, c ) is κ -disconnected if there is a sequence( a ι ) ι<κ ⊆ B of nonzero pairwise orthogonal clopen elements such that W ι<κ a ι = 1. Lemma 6.5. If X is an infinite Hausdorff Baire space, then Baire ( X ) is ω -disconnected.Proof. First suppose that X has no limit points. Then, every point is isolated, andsince X is infinte we can choose a sequence of distinct points ( x i ) i<ω . Then eachsingleton { x i } is open, hence clopen and non-meager. Let a = [ X \ { x i } ≤ i<ω ] andfor i > a i = [ { x i } ]. Then ( a i ) i<ω is a sequence of nonzero pairwise orthogonalclopens of Baire ( X ), witnessing that X is ω -disconnected.Next suppose that X has a limit point, say x ∗ . We define a sequence of points( x i ) i<ω and two sequences of open sets ( U i ) i ≤ n and ( V i ) i ≤ n as follows.To begin, choose x = x ∗ , and let U , V be disjoint open neighborhoods of x , x ∗ , respectively. For the inductive step, suppose that ( x i ) i ≤ n , ( U i ) i ≤ n , ( V i ) i ≤ n are already chosen and satisfy the following conditions:(1) for all i ≤ n , we have x i ∈ U i , x ∗ ∈ V i , and U i , V i are open;(2) if i < j ≤ n , then U i ∩ U j = ∅ , and(3) if i ≤ j ≤ n , then U i ∩ V j = ∅ .Since x ∗ is a limit point, choose x n +1 ∈ V n \ { x ∗ } . Let U, V be disjoint neighbor-hoods of x n +1 , x ∗ , respectively, and define U n +1 = U ∩ V n , V n +1 = V ∩ V n . It isnot hard to check that the sequences ( x i ) i ≤ n +1 , ( U i ) i ≤ n +1 , ( V i ) i ≤ n +1 satisfy all thedesired properties.Once we have constructed ( U i ) i<ω , we let a = [ X \ S ≤ n<ω U n ] and for i > a i = [ U i ]. Then ( a i ) i<ω is the desired sequence of nonzero pairwise orthogonalclopens of Baire ( X ). (cid:3) Remark 6.6. By Lemma 6.5, the Baire algebra of the Cantor space is ω -dis-connected. In contrast, the Cantor space, while disconnected, is not ω -disconnectedbecause, by compactness, any partition into disjoint open sets must be finite. Moregenerally, no compact space can be ω -disconnected. Remark 6.7. Let X be an infinite separable Hausdorff Baire space. Then Baire ( X )is ω -disconnected by Lemma 6.5. On the other hand, Baire ( X ) is not κ -disconnec-ted for any cardinal κ > ω . For, suppose that ( a i ) i<κ ⊆ Baire ( X ) are pairwiseorthogonal and clopen. Then for each i < κ , there is an open set A i ⊆ X with a i = [ A i ]. We must have that A i ∩ A j = ∅ if i = j , for otherwise their intersection,being open, would be non-meager. It follows that κ ≤ ω as no separable spaceadmits an uncountable collection of disjoint open subsets. Theorem 6.8. Let X be a Baire space and W an S5 -frame with the upper clustersize λ and the number of clusters κ . If Baire ( X ) is κ -disconnected and λ -resolvable,then there is an embedding h : Kur ( W ) → Baire ( X ) .Proof. Let ( C ι ) ι<κ enumerate the clusters of W and let λ ι = | C ι | . Enumerate each C ι by ( w ιν ) ν<λ ι . Since Baire ( X ) is κ -disconnected, there are pairwise orthogonalelements ([ U ι ]) ι<κ such that each U ι is open and F ι<κ [ U ι ] = [ X ]. Because X isa Baire space, the U ι must be pairwise disjoint as their intersection is open butmeager. By Lemma 4.7, for each ι < κ and ν < λ ι , there are A ιν ⊆ U ι that Baire λ ι -resolve U ι .Define a partial function f : X → W by f ( x ) = w if there exist ι < κ , ν < λ ι such that x ∈ A ιν and w = w ιν . We show that f is a Baire map. It is continuoussince if V ⊆ W is open, then V is a union of clusters S ι ∈ I C ι , hence f − ( V ) = S ι ∈ I f − ( C ι ) = S ι ∈ I U ι . Therefore, f is Baire-continuous. It is Baire-open since if V ⊆ X is open, M ⊆ X is meager, and w ∈ f ( V \ M ), then w = w ιν for some ι , ν .Therefore, for any ν ′ < λ ι , we have that A ιν ′ ∩ V is non-meager, so ( A ιν ′ ∩ V ) \ M isnonempty, and hence w ιν ′ ∈ f ( V \ M ). Since ν ′ was arbitrary, C ι ⊆ f ( V \ M ), andsince w was arbitrary, f ( V \ M ) is open. The map f is defined almost everywheresince [ X ] = F ι<κ [ U ι ], so X \ S ι<κ U ι is meager. It is trivially proper since theonly meager subset of W is the empty set. Finally, it is exact since f − ( w ιν ) = A ιν ,which is non-meager. Thus, h : Kur ( W ) → Baire ( X ) given by h ( B ) = [ f − ( B )] isan embedding by Lemma 5.7. (cid:3) Theorem 6.9. If X is ω -resolvable, then S5U is sound and strongly complete for Baire ( X ) .Proof. Soundness follows from Baire ( X ) being a monadic algebra. For complete-ness, if Γ S5U ϕ , then by Theorem 6.1, there is a countable S5U -frame W anda valuation J · K W such that Γ = W ϕ . By Theorem 6.8, there is an embedding f : Kur ( W ) → Baire ( X ). Thus, Γ = Baire ( X ) ϕ . (cid:3) Corollary 6.10. Let X be a crowded space which is either completely metrizableand of cardinality continuum or locally compact Hausdorff. Then S5U is sound andstrongly complete for Baire ( X ) . In particular, S5U is sound and strongly completefor Baire ( R ) and Baire ( C ) , where C is the Cantor space. Corollary 6.10 can be viewed as an analogue of the McKinsey-Tarski theoremfor S5U . Remark 6.11. Corollary 6.10 does not extend to uncountable languages in viewof Remark 6.7, as the collection {∃ (cid:3) p ι } ι<κ ∪ {¬∃ ( p ι ∧ p ς ) } ι<ς<κ cannot be satisfied on e.g. the real line if κ > ω . HE BAIRE CLOSURE AND ITS LOGIC 21 Concluding remarks We have shown that the Baire algebra of any topological space is a closure alge-bra, and that the logic of these algebras is S5 , providing variants of the celebratedMcKinsey-Tarski theorem for the logic S5 . We have identified Baire resolvability asa sufficient condition for strong completeness. For such spaces, we have also con-structed subalgebras of the Baire algebra whose logic is S5 n for any finite n . Given acardinal λ , strong completeness also holds for variants of S5 with λ -many variablesprovided that the space is Baire λ -resolvable. We have also shown that if the Bairealgebra is ω -disconnected, then strong completeness extends to S5U . Finally, wehave shown that crowded spaces which are either complete continuum-sized metricor locally compact Hausdorff enjoy the above properties, leading to a large class ofconcrete examples of spaces for which S5 and S5U are strongly complete.This work follows [11, 21] in studying point-free semantics for modal logic basedon quotients of the powerset algebras of topological spaces. There are many other σ -ideals that may be used to define similar quotients, e.g. the σ -ideal of countablesubsets. 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Tsao-Chen, Algebraic postulates and a geometric interpretation for the Lewis calculus ofstrict implication , Bulletin of the American Mathematical Society (1938), 737–744. New Mexico State University Email address : [email protected] Ghent University Email address ::