Abstract
In this paper we consider the topological interpretations of , the classical logic extended by a “box” operator interpreted as interior. We present extensions of S4 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.
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 logic S4 is complete in any metric separable dense-in-itself space, for example, . The original proof of McKinsey and Tarski was quite tedious and technical; simpler, shorter, more geometric proofs were given in [2–5]. It follows that S4 is sound and complete over all topological spaces; that is, a formula is derivable from the axiom system of S4 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 from S4.
In Section 2 we recall all necessary definitions, and in Section 3 we consider two axioms, denoted by and , each not derivable from S4 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. Let be a CW-complex. Its quotient space is a topological space whose points are in one-to-one correspondence with cells of , and a subset of is open if and only if the union of the corresponding cells is open in .
Definition 2. If is a CW-complex, its cell is called a top cell if it is not in the boundary of any other cell.
Remark 3. A single point in is open if and only if it corresponds to a top cell.
Definition 4. Let be any nonempty set and . Collection of subsets of is called the particular point topology on ([6, p. 44]).
Remark 5. Since, for any particular point topological space with the particular point , any nonempty open subset includes , it follows that the only closed subset that includes is .
Definition 6. Let be any nonempty set and . Collection of subsets of is called the excluded point topology on ([6, p. 47]).
Remark 7. Since, for any excluded point topological space with the excluded point , every proper open subset of excludes , it follows that every nonempty closed set includes .
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 (real projective plane) (see Figure 1).
Example 10. More generally, for each , let with topology (see Figure 2).For each , with topology is the quotient space of the standard CW-complex for .
Remark 11. Topology is a subtopology of both a particular point topology (with the particular point being ) and an excluded point topology (with the excluded point being ).
2.2. Modal Logic Language and Its Interpretation in Topological Spaces
Definition 12. Let be a modal logic language consisting of propositional variables, conjunction , disjunction , negation , and modality . Modality is defined as the dual to ; that is, .
Definition 13. S4 denotes the subset of consisting of all formulas that can be derived from the following axioms: (i)All axioms of the classical propositional logic(ii)(iii)(iv) using the following rules: (i) (Modus ponens)(ii) (necessitation).
Definition 14. A topological model of is a pair , where(1) is a topological space(2) is a valuation function mapping formulas in to subsets of . It assigns each propositional variable an arbitrary subset and satisfies the following conditions for every : where is the topological interior operator.
Remark 15. Note that the modality then maps to the topological closure operator; that is, .
Definition 16. Let be any formula. One says that is valid in a topological space if, for any topological model , one has .
Theorem 17 (topological completeness of S4, [1–5]). For any formula , the following two statements are equivalent: (i) is derivable in .(ii) is valid in all topological spaces.
3. Axioms and
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):
Theorem 18. Both axioms and are valid in any particular point topological space.
Proof. Let be a particular point topological space with particular point , and let be any subset of .
Case 1 (). Then is open. So . Since the only closed set that includes is , . Therefore .
Also, since the only closed set that includes is , . Then . So .
Case 2 (). Then . The rest is similar to case , with the roles of and switched.
Theorem 19. (1) Axiom is valid in any excluded point topological space.
(2) Axiom is valid in an excluded point topological space if and only if the space has only 1 or 2 points.
Proof. Let be an excluded point topological space with excluded point , and let be any subset of .
Case 1 (). Then .
Case 2 (). Then , so .
Case 3 (, ). Then .
Case 4 (, ). Then , ; thus .
Case 5 ( and ). Then and .
Case 6 (, , and ). Then and Thus this case is similar to case , with the roles of and switched.
In each of these cases, we have An excluded point topological space with only one or two points is also a particular point topological space; thus axiom is valid by Theorem 18. If there are at least three points, let be the excluded point and let be any other point. Consider . Then and , so .
Remark 20. It is known (see, e.g., [7]) that axioms and are independent from each other and S4. Theorem 19 provides another proof of independency of from S4 + Axiom , traditionally denoted by S4.1.
Theorem 21. (1) Axiom is valid in the quotient space of any finite CW-complex.
(2) Axiom 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. Let be any finite CW-complex, and let be any point. Then is in the closure of some point corresponding to a top cell, so is open.
Case 1 (). Then .
Case 2 (). Then , so .
If corresponds to a top cell, then its entire connected component is in . Since each connected component of a finite CW-complex contains a top cell and the point in corresponding to this top cell is either in or in , .
(⇒) 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. Let and be points in the quotient space corresponding to such two top cells and let be the point corresponding to a cell in the intersection of their closures. Consider a validation mapping such that and . Since corresponds to a top cell, it is not in the boundary of any other point; therefore , so . Then ; therefore . Since , in a similar way, we have . Thus .
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The authors are very grateful to Fred Cohen for suggesting the study of quotient spaces of CW-complexes and to the College of Science and Mathematics of the California State University, Fresno for supporting this work.