# Modal Logic Axioms Valid in Quotient Spaces of Finite CW-Complexes and Other Families of Topological Spaces

Authors: Maria Nogin; Bing Xu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2016)

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## Abstract

In this paper we consider the topological interpretations ofL□, the classical logic extended by a “box” operator □ interpreted as interior. We present extensions ofS4 that are sound over some families of topological spaces, including particular point topological spaces, excluded point topological spaces, and quotient spaces of finite CW-complexes.


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## Body

## 1. Introduction

It was shown by McKinsey and Tarski in [1] that every finite well-connected topological space is an open image of a metric separable dense-in-itself space. This implies that the modal logicS4 is complete in any metric separable dense-in-itself space, for example, Rn. The original proof of McKinsey and Tarski was quite tedious and technical; simpler, shorter, more geometric proofs were given in [2–5]. It follows thatS4 is sound and complete over all topological spaces; that is, a formula is derivable from the axiom system ofS4 if and only if it is valid in all topological spaces. Note that if a formula is valid in some topological space, it does not follow that this formula is derivable fromS4.In Section2 we recall all necessary definitions, and in Section 3 we consider two axioms, denoted by M and G, each not derivable fromS4 and the other one, and for each of them we give necessary and sufficient conditions under which it is valid in a quotient space of a finite CW-complex, a particular point topological space, and an excluded point topological space.


## 2. Preliminaries

### 2.1. CW-Complexes and Their Quotient Spaces, Particular Point Topological Spaces, and Excluded Point Topological Spaces

Definition 1.
LetX be a CW-complex. Its quotient space Q(X) is a topological space whose points are in one-to-one correspondence with cells of X, and a subset of Q(X) is open if and only if the union of the corresponding cells is open in X.Definition 2.
IfX is a CW-complex, its cell c is called a top cell if it is not in the boundary of any other cell.Remark 3.
A single point inQ(X) is open if and only if it corresponds to a top cell.Definition 4.
LetX be any nonempty set and p∈X. Collection Tp={S⊆X∣p∈SorS=∅} of subsets of X is called theparticular point topology on X ([6, p. 44]).Remark 5.
Since, for any particular point topological spaceX with the particular point p∈X, any nonempty open subset includes p, it follows that the only closed subset that includes p is X.Definition 6.
LetX be any nonempty set and e∈X. Collection Te={S⊆X∣e∉SorS=X} of subsets of X is called theexcluded point topology on X ([6, p. 47]).Remark 7.
Since, for any excluded point topological spaceX with the excluded point e∈X, every proper open subset of X excludes e, it follows that every nonempty closed set includes e.Example 8.
The quotient space of a bouquet of spheres (of any positive dimension) has excluded point topology.Example 9.
Sierpiński space, a topological space with two points only one of which is open ([6, p. 44]), is a topological space with both particular point and excluded point topology. It is the quotient space of the standard CW-complex for RP1 (real projective plane) (see Figure 1).Figure 1Example 10.
More generally, for eachn∈N, let Xn={x1,x2,…,xn} with topology (see Figure 2).(1)Ts=S⊆X∣S=∅orS=x1,x2,…,xkfor1≤k≤n.
For eachn∈N, Xn with topology Ts is the quotient space of the standard CW-complex for RPn-1.Figure 2Remark 11.
TopologyTs is a subtopology of both a particular point topology (with the particular point being x1) and an excluded point topology (with the excluded point being xn).


### 2.2. Modal Logic LanguageL□ and Its Interpretation in Topological Spaces

Definition 12.
LetL□ be a modal logic language consisting of propositional variables, conjunction ∧, disjunction ∨, negation ¬, and modality □. Modality ◊ is defined as the dual to □; that is, ◊P=¬□¬P.Definition 13.
S4 denotes the subset of L□ consisting of all formulas that can be derived from the following axioms: (i)
All axioms of the classical propositional logic(ii)
□(A→B)→(□A→□B)(iii)
□A→A(iv)
□A→□□A using the following rules: (i)
A,A→B/B (Modus ponens)(ii)
A/□A (necessitation).Definition 14.
Atopological model of L□ is a pair X,·, where(1)
X is a topological space(2)
· is a valuation function mapping formulas in L□ to subsets of X. It assigns each propositional variable P an arbitrary subset P⊆X and satisfies the following conditions for every F,G∈L□:(2)F∧G=F∩G,F∨G=F∪G,¬F=FC=X∖F,□F=intF, where int(·) is the topological interior operator.Remark 15.
Note that the modality◊ then maps to the topological closure operator; that is, ◊F=cl(F).Definition 16.
LetF∈L□ be any formula. One says that F isvalid in a topological space X if, for any topological model X,·, one has F=X.Theorem 17 (topological completeness ofS4, [1–5]).
For any formulaF∈L□, the following two statements are equivalent: (i)
F is derivable in S4.(ii)
F is valid in all topological spaces.


## 2.1. CW-Complexes and Their Quotient Spaces, Particular Point Topological Spaces, and Excluded Point Topological Spaces

Definition 1.
LetX be a CW-complex. Its quotient space Q(X) is a topological space whose points are in one-to-one correspondence with cells of X, and a subset of Q(X) is open if and only if the union of the corresponding cells is open in X.Definition 2.
IfX is a CW-complex, its cell c is called a top cell if it is not in the boundary of any other cell.Remark 3.
A single point inQ(X) is open if and only if it corresponds to a top cell.Definition 4.
LetX be any nonempty set and p∈X. Collection Tp={S⊆X∣p∈SorS=∅} of subsets of X is called theparticular point topology on X ([6, p. 44]).Remark 5.
Since, for any particular point topological spaceX with the particular point p∈X, any nonempty open subset includes p, it follows that the only closed subset that includes p is X.Definition 6.
LetX be any nonempty set and e∈X. Collection Te={S⊆X∣e∉SorS=X} of subsets of X is called theexcluded point topology on X ([6, p. 47]).Remark 7.
Since, for any excluded point topological spaceX with the excluded point e∈X, every proper open subset of X excludes e, it follows that every nonempty closed set includes e.Example 8.
The quotient space of a bouquet of spheres (of any positive dimension) has excluded point topology.Example 9.
Sierpiński space, a topological space with two points only one of which is open ([6, p. 44]), is a topological space with both particular point and excluded point topology. It is the quotient space of the standard CW-complex for RP1 (real projective plane) (see Figure 1).Figure 1Example 10.
More generally, for eachn∈N, let Xn={x1,x2,…,xn} with topology (see Figure 2).(1)Ts=S⊆X∣S=∅orS=x1,x2,…,xkfor1≤k≤n.
For eachn∈N, Xn with topology Ts is the quotient space of the standard CW-complex for RPn-1.Figure 2Remark 11.
TopologyTs is a subtopology of both a particular point topology (with the particular point being x1) and an excluded point topology (with the excluded point being xn).


## 2.2. Modal Logic LanguageL□ and Its Interpretation in Topological Spaces

Definition 12.
LetL□ be a modal logic language consisting of propositional variables, conjunction ∧, disjunction ∨, negation ¬, and modality □. Modality ◊ is defined as the dual to □; that is, ◊P=¬□¬P.Definition 13.
S4 denotes the subset of L□ consisting of all formulas that can be derived from the following axioms: (i)
All axioms of the classical propositional logic(ii)
□(A→B)→(□A→□B)(iii)
□A→A(iv)
□A→□□A using the following rules: (i)
A,A→B/B (Modus ponens)(ii)
A/□A (necessitation).Definition 14.
Atopological model of L□ is a pair X,·, where(1)
X is a topological space(2)
· is a valuation function mapping formulas in L□ to subsets of X. It assigns each propositional variable P an arbitrary subset P⊆X and satisfies the following conditions for every F,G∈L□:(2)F∧G=F∩G,F∨G=F∪G,¬F=FC=X∖F,□F=intF, where int(·) is the topological interior operator.Remark 15.
Note that the modality◊ then maps to the topological closure operator; that is, ◊F=cl(F).Definition 16.
LetF∈L□ be any formula. One says that F isvalid in a topological space X if, for any topological model X,·, one has F=X.Theorem 17 (topological completeness ofS4, [1–5]).
For any formulaF∈L□, the following two statements are equivalent: (i)
F is derivable in S4.(ii)
F is valid in all topological spaces.


## 3. AxiomsM and G

Consider the following two axioms (the first form of each axiom is the one appearing most often in the literature, but the second one will be more convenient for us):(3)M:□◊P⟶◊□¬P≡◊□P∨◊□¬P,G:◊□P⟶□◊¬P≡□◊P∨□◊¬P.Theorem 18.
Both axiomsM and G are valid in any particular point topological space.Proof.
LetX be a particular point topological space with particular point p, and let P=A be any subset of X.
Case  1 (p∈A). Then A is open. So int(A)=A. Since the only closed set that includes p is X, cl(int(A))=cl(A)=X. Therefore (◊□P)∨(◊□¬P)=cl(int(A))∪cl(int(AC))=X.
Also, since the only closed set that includesp is X, cl(A)=X. Then int(cl(A))=int(X)=X. So (□◊P)∨(□◊¬P)=int(cl(A))∪int(cl(AC))=X.
Case  2 (p∉A). Then p∈AC. The rest is similar to case 1, with the roles of A and AC switched.Theorem 19.
(1) AxiomM is valid in any excluded point topological space.
(2) AxiomG is valid in an excluded point topological space if and only if the space has only 1 or 2 points.Proof.
( 1 ) Let X be an excluded point topological space with excluded point e, and let P=A be any subset of X.
Case  1 (A=X). Then cl(int(A))=cl(int(X))=X.
Case  2 (A=∅). Then AC=X, so cl(int(AC))=cl(int(X))=X.
Case  3 (A=X∖{e}, A≠∅). Then cl(int(A))=cl(A)=X.
Case  4 (A={e}, A≠X). Then AC=X∖{e}, AC≠∅; thus cl(int(AC))=X.
Case  5 (e∈A⊂X and A≠{e}). Then cl(int(A))=cl(A∖{e})=A and cl(int(AC))=cl(AC)=AC∪{e}.
Case  6 (e∉A, A≠∅, and A≠X∖{e}). Then e∈AC⊂X and AC≠{e}. Thus this case is similar to case 5, with the roles of A and AC switched.
In each of these cases, we have(4)◊□P∨◊□¬P=clintA∪clintAC=X.
( 2 ) An excluded point topological space with only one or two points is also a particular point topological space; thus axiom G is valid by Theorem 18. If there are at least three points, let e be the excluded point and let a be any other point. Consider A={a,e}. Then int(cl(A))=int(cl({a,e}))=int({a,e})={a} and int(cl(AC))=int(cl(X∖{a,e}))=int(X∖{a})=X∖{a,e}, so (□◊P)∨(□◊¬P)=int(cl(A))∪int(cl(AC))=X∖{e}≠X.Remark 20.
It is known (see, e.g., [7]) that axioms M and G are independent from each other andS4. Theorem 19 provides another proof of independency of G fromS4+ Axiom M, traditionally denoted byS4.1.Theorem 21.
(1) AxiomM is valid in the quotient space of any finite CW-complex.
(2) AxiomG is valid in the quotient space of a finite CW-complex iff each connected component of the CW-complex has a unique top cell.Proof.
( 1 ) Let X be any finite CW-complex, and let c∈Q(X) be any point. Then c is in the closure of some point t corresponding to a top cell, so t is open.
Case  1 (t∈P). Then c∈cl({t})=cl(int({t}))⊆cl(int(P)).
Case  2 (t∉P). Then t∈PC, so c∈cl(int(PC)).
( 2 )   ( ⇐ ) If t corresponds to a top cell, then its entire connected component is in int(cl({t})). Since each connected component of a finite CW-complex contains a top cell and the point in Q(X) corresponding to this top cell is either in P or in PC, (□◊P)∨(□◊¬P)=X.
(⇒) If a connected component of a finite CW-complex contains at least two top cells, then there two top cells whose closures have nonempty intersection, so the intersection contains at least one cell. Lett1 and t2 be points in the quotient space corresponding to such two top cells and let c be the point corresponding to a cell in the intersection of their closures. Consider a validation mapping such that t1∈P and t2∉P. Since t2 corresponds to a top cell, it is not in the boundary of any other point; therefore t2∉cl(P), so t2∈(cl(P))C. Then c∈cl({t2})⊆cl((cl(P))C); therefore c∉(cl((cl(P))C))C=¬◊¬◊P=□◊P. Since t1∉cl(¬P), in a similar way, we have c∉□◊¬P. Thus (□◊P)∨(□◊¬P)≠X.



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