Abstract
In this chapter we study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, \(T_1\)-spaces, dense-in-themselves spaces, a zero-dimensional dense-in-itself separable metric space, \(\mathbf R^n~ (n\ge 2).\) We also discuss the correlation between languages with different combinations of the topological, derivational, universal and difference modalities in terms of definability.
In memory of Leo Esakia
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Notes
- 1.
The early works of the second author in this field were greatly influenced by Leo Esakia.
- 2.
Some other kinds of topomodal logics arise when we deal with topological spaces with additional structures, e.g. spaces with two topologies, spaces with a homeomorphism etc. (cf. [3]).
- 3.
For the 1-modal case this lemma has been known as folklore since the 1970s; the second author learned it from Leo Esakia in 1975.
- 4.
Sometimes we neglect this difference.
- 5.
There is no common notation for this operation; some authors use \(\varvec{\uptau }\).
- 6.
So we extend the definitions of the d-truth or the c-truth by adding the item for \([\forall ]\) or \([\ne ]\).
- 7.
Shehtman [37] contains a stronger claim: \(\mathbf {Lc_\forall }(\mathfrak X)=\mathbf {S4U} + AC\) for any connected dense-in-itself separable metric \(\mathfrak X\). However, recently we found a gap in the proof of Lemma 17 from that paper. Now we state the main result only for the case \({\mathfrak X}=\mathbf R^n\); a proof can be obtained by applying the methods of the present chapter, but we are planning to publish it separately.
- 8.
- 9.
\(f_1\cup f_2\) is the map \(f\) such that \(f|X_i=f_i\); similarly for \(f_1\cup f_2\cup g\) (Fig. 11.2).
- 10.
\(|\ldots |\) denotes the cardinality.
- 11.
Basic frames were defined in Sect. 11.4.
- 12.
Recall that \(R^\prime \) is the transitive closure of \({\underline{R}}\), \(R_D'=\underline{R_D}\).
- 13.
In this chapter, as well as in [9, ‘countable’ means ‘of cardinality at most \(\aleph _0\)’.
- 14.
In [31] neighbourhoods are assumed to be open, but this does not matter here since every neighbourhood contains an open neighbourhood.
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Acknowledgments
We would like to thank the referee who has helped us to improve the first version of the manuscript. The work on this chapter was supported by RFBR grants 11-01-00281-a, 11-01-00958-a, 11-01-93107-CNRS-a and the Russian President’s grant NSh-5593.2012.1.
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Appendix
Appendix
Finally let us give technical details of the proofs of Propositions 11, 19.
Proposition 11 Let \(\mathfrak X\) be a dense-in-itself separable metric space, \(B\subset X\) a closed nowhere dense set. Then there exists a d-morphism \(g: \mathfrak X\twoheadrightarrow ^d \Phi _{ml}\) with the following properties:
-
(1)
\(B \subseteq g^{-1}(b_1)\);
-
(2)
every \(g^{-1}(a_i)\) \((for i\le l)\) is a union of a set \(\alpha _i\) of disjoint open balls, which is dense at any point of \(g^{-1}(\{ b_1,\ldots ,b_m\})\).
The frame \(\Phi _{ml}\) is shown in Fig. 11.1.
Proof
Let \(X_1,\dots , X_n,\dots \) be a countable base of \(\mathfrak X\) consisting of open balls. We construct sets \(A_{ik}, ~B_{jk}\) for \(1\le i\le l, ~1\le j\le m, ~k\in \omega \), with the following properties:
-
(1)
\(A_{ik}\) is the union of a finite set \(\alpha _{ik}\) of nonempty open balls whose closures are disjoint;
-
(2)
\(\mathbf{C}A_{ik} \cap \mathbf{C}A_{i'k} = \varnothing \) for \(i\not = i'\);
-
(3)
\(\alpha _{ik} \subseteq \alpha _{i,k+1};~A_{ik} \subseteq A_{i,k+1}\);
-
(4)
\(B_{jk}\) is finite;
-
(5)
\(B_{jk} \subseteq B_{j,k+1};\)
-
(6)
\(A_{ik} \cap B_{jk} = \varnothing ;\)
-
(7)
\(X_{k+1} \subseteq \bigcup \limits _{i=1}^l A_{ik}\Rightarrow \alpha _{i,k+1} = \alpha _{ik}, ~B_{j,k+1} = B_{jk};\)
-
(8)
if \(X_{k+1}\not \subseteq \bigcup \limits _{i=1}^l A_{ik}\), there are closed nontrivial balls \(P_1,\dots ,P_l\) such that for any \(i\), \(j\)
$$P_i \subseteq X_{k+1}-A_{ik},~ \alpha _{i,k+1} = \alpha _{ik} \cup \{ \mathbf{I}P_i\},~ (B_{j,k+1} - B_{jk}) \cap X_{k+1}\not = \varnothing ; $$ -
(9)
\(A_{ik} \subseteq X-B\);
-
(10)
\(B_{jk} \subseteq X-B\);
-
(11)
\(j\not = j'\Rightarrow B_{j'k} \cap B_{jk} = \varnothing \) .
We carry out both the construction and the proof by induction on \(k\).
Let \(k=0\); \((X-B)\) is infinite since it is nonempty and open in a dense-in-itself \(\mathfrak X\). Take distinct points \(v_1,\dots ,v_l \not \in B\) and disjoint closed nontrivial balls \( Z_1,\dots ,Z_l \subset X-B\) with centers at \(v_1,\dots ,v_l\), respectively (see Fig.11.6).
Put
then \(Z_i = \mathbf{C}A_{i0}\). As above, since \((X-B) -\bigcup \limits ^l_{i=1}Z_i\) is nonempty and open, it is infinite. Pick distinct \(w_1,\dots ,w_m\in X-B\) and put \(B_{j0}: = \{ w_j\}\). Then the required properties hold for \(k=0\).
At the induction step we construct \(A_{i,k+1}, B_{j,k+1}\). Put \(Y_k :=\bigcup \limits ^l_{i=1}A_{ik}\) and consider two cases.
(a) \(X_{k+1}\subseteq Y_k\). Then put:
(b) \(X_{k+1}\not \subseteq Y_k\). Then \(X_{k+1} \not \subseteq \mathbf{C}Y_k\). Indeed, \(X_{k+1} \subseteq \mathbf{C}Y_k\) implies \(X_{k+1} \subseteq \mathbf{I}\mathbf{C}Y_k=Y_k\) since \(X_{k+1}\) is open and by (1) and (2). So we put
Since \((X_{k+1} - \mathbf{C}Y_k )\) is nonempty and open and every \(B_{jk}\) is finite by (4), \(W_0\) is also open and nonempty (by the density of \(\mathfrak X\)). By the assumption of Proposition 11, \(B\) is closed, and thus \(W\) is open.
\(W\) is also nonempty. Otherwise \(W_0 \subseteq B\), and then \(W_0 \subseteq \mathbf{I}B=\varnothing \) (since \(B\) is nowhere dense by the assumption of Proposition 11).
Now we argue similarly to the case \(k=0\). Take disjoint closed nontrivial balls \(P_1,\dots ,P_l\subset W\). Then \(W - \bigcup \limits ^l_{i=1}P_i\) is infinite, so we pick distinct \(b_{1,k+1},\dots , b_{m,k+1}\) in this set and put
In case (a) all the required properties hold for \((k+1)\) by the construction.
In case (b) we have to check only (1), (2), (6), (8)–(11).
(8) holds since by construction we have
(1): From IH it is clear that \(\alpha _{i,k+1}\) is a finite set of open balls and their closures are disjoint; note that \(P_i \cap \mathbf{C}A_{ik} = \varnothing \) since \(P_i \subseteq W \subseteq -\mathbf{C}A_{ik}.\)
(2): We have
by IH and by the construction; note that \(P_i, P_i' \subseteq W \subseteq -\mathbf{C}Y_k\).
(6): We have
by IH and since \(b_{j,k+1}\not \in P_i,~ b_{j,k+1}\in W \subseteq X-Y_k\), \(P_i \subset W \subseteq X-B_{jk}.\)
(9): We have \(A_{i,k+1} = A_{ik} \cup \mathbf{I}P_i \subseteq -B\) since \(A_{ik} \subseteq -B\) by IH, and \(P_i \subset W \subseteq -B\) by the construction.
Likewise, (10) follows from \(B_{jk} \subseteq -B\) and \(b_{j,k+1}\in W \subseteq -B\).
To check (11), assume \(j \not = j'\). We have \(B_{j',k+1} \cap B_{j,k+1} = B_{j'k} \cap B_{jk}\) since \(b_{j^\prime ,k+1} \not = b_{j,k+1} , ~ b_{j,k+1}\in W \subseteq -B_{j^\prime k}\) and \(b_{j^\prime ,k+1}\in W\subseteq -B_{jk}\). Then apply IH.
Therefore, the required sets \(A_{ik}, B_{jk}\) are constructed. Now put
and define a map \(g: X\longrightarrow \Phi _{ml}\) as follows:
By (2), (3), (5), (6), (11), \(g\) is well defined; by (9), (10), \(B \subseteq g^{-1}(b_1)\).
To prove that \(g\) is a d-morphism, we check some other properties.
Indeed, take an arbitrary \(x\not \in \bigcup \limits ^l_{i=1}A_i \) and show that \(x\in \mathbf{d}B_j\), i.e.,
for any neighbourhood \(U\) of \(x\). First assume that \(x\not \in B_j\). Take a basic open \(X_{k+1}\) such that \(x\in X_{k+1} \subseteq U\). Then \(X_{k+1}\not \subseteq \bigcup \limits ^l_{i=1}A_i\), and (8) implies \(B_{j,k+1} \cap X_{k+1} \not = \varnothing .\) Thus \(B_j \cap U \not = \varnothing \). So we obtain (13).
Suppose \(x\in B_j\); then \(x\in B_{jk}\) for some \(k\). Since \(\mathfrak X\) is dense-in-itself and \(\left\{ X_1,\,X_2,\dots \right\} \) is its open base, \(\left\{ X_{s+1}\,\left| \,s\ge k \right. \right\} \) is also an open base (note that every ball in \(\mathfrak X\) contains a smaller ball). So \(x\in X_{s+1}\subseteq U\) for some \(s\ge k\). Since \(x\not \in \bigcup \limits ^l_{i=1}A_i \), we have \(X_{s+1}\not \subseteq \bigcup \limits ^l_{i=1}A_i\), and so \((B_{j,s+1}-B_{js}) \cap X_{s+1} \not = \varnothing \) by (8); thus \((B_j-B_{js}) \cap U \not = \varnothing \). Now \(x\in B_{jk} \subseteq B_{js}\) implies (13).
Indeed, \(B_j\subseteq -A_i\) by (3), (5), (6). So \(\mathbf{d}B_j \subseteq \mathbf{d}(-A_i) \subseteq -A_i\) since \(A_i \) is open.
Similarly we obtain
Also note that
since \(A_i\) is open, \(\mathfrak X\) is dense-in-itself. As in (12) we have
To conclude that \(g\) is a d-morphism, note that
and so by (15), (16), (17)
and by (12), (14), (15)
Proposition 19 For a finite rooted \(\mathbf {DT_1CK}\) -frame \(F = (W, R, R_D)\) and \(R\) -reflexive points \(w', w'' \in W\), the following holds.
-
(a)
If \(X=\left\{ x\in \mathbf{R}^n\mid ||x|| \le r \right\} \), \(n \ge 2\), then there exists \(f: X \twoheadrightarrow ^{dd}F\) such that \(f(\partial X) = \left\{ w'\right\} ;\)
-
(b)
If \(0\le r_1<r_2\) and
$$\begin{aligned} X&= \left\{ x\in \mathbf{R}^n\,\left| \,r_1\le ||x||\le r_2 \right. \right\} , \\ Y'&=\left\{ x\in \mathbf{R}^n\,\left| \,||x||=r_1 \right. \right\} ,\ Y'' =\left\{ x\in \mathbf{R}^n\,\left| \,||x||=r_2 \right. \right\} , \end{aligned}$$then there exists \(f: X \twoheadrightarrow ^{dd}F\) such that \(f(Y') = \left\{ w'\right\} \), \(f(Y'') = \left\{ w''\right\} .\)
Proof By induction on \(\left| W\right| \). Let us prove (a) first. There are five cases:
(a1) \(W = R(b)\) (and hence \(b R b\)) and \(b = w'\). Then there exists \(f:X\twoheadrightarrow ^d (W,R)\). Indeed, let \(C\) be the cluster of \(b\) (as a subframe of \((W,R)\)). Then \((W,R)=C\) or \((W,R)=C\cup F_1\cup \ldots \cup F_l\), where the \(F_i\) are generated by the successors of \(C\). If \((W,R)=C\), we apply Proposition 11; otherwise we apply Lemma 15 and IH.
By Proposition 8 it follows that \(R_D\) is universal. And so by Proposition 12(3) \(f\) is a dd-morphism.
(a2) \(W = R(b)\) and not \(w'Rb\). We may assume that \(r=3\). Put
By the case (a1), there is \(f_1: X_1 \twoheadrightarrow ^{dd}F\) with \(f_1(\partial X_1) = \left\{ b\right\} \). Let \(C\) be a maximal cluster in \(R(w')\). By Proposition 11 there is \(g: \mathbf{I}Y \twoheadrightarrow ^d C\). Since \(R(w')\ne W\), we can apply IH to the frame \(F':=F^{w'}_{\forall }\) and construct a dd-morphism \(f_2: X_2 \twoheadrightarrow ^{dd}F'\) with \(f_2(\partial X_2) = \left\{ w'\right\} \). Now since \(f_i(\partial X_i)\subseteq R^{-1}(C)\), the Glueing lemma 16 is applicable. Thus \(f:X\twoheadrightarrow ^d F\) for \(f:=f_1\cup f_2\cup g\) [See Fig. 11.7, Case (a2)]. Note that \(\partial X\subset \partial X_2\), so \(f(\partial X)=f_2(\partial X)=\{w'\}\).
As in case (a1), \(f\) is a dd-morphism by Proposition 12.
(a3) \((W,R)\) is not rooted. By Lemma 33 there is a global path \(\alpha \) in \(F\) with a single occurrence of every \(R_D\)-irreflexive point. We may assume that \( \alpha = b_0 c_0 b_1 c_1\ldots c_{m-1} b_m\), \(b_m = w'\) and for any \(i<m\), \(c_i \in C_i \subseteq R(b_i) \cap R(b_{i+1})\), where \(C_i\) is an \(R\)-maximal cluster. Such a path is called reduced. For \(0 \le j \le m\) we put \(F_j := F|\overline{R}(b_j)\).
Since \((W,R)\) is not rooted, each \(F_j\) is of smaller size than \(F\), so we can apply the induction hypothesis to \(F_j\). We may assume that
Then put
By IH and Proposition 11 there exist
One can check that \(f:X\twoheadrightarrow ^{dd} F\) for \(f:=\bigcup \limits _{j=0}^{2m}f_j\) (Fig. 11.7).
(a4) \(W = \overline{R}(b)\), \(\lnot b R_D b\) (and so \(\lnot b R b\)). We may assume that
Then similar to case (a3) put
Consider the frame \(F' {:}= F|W'\), where \(W' = W - \left\{ b\right\} \). Note that \(w'\in W'\) since \(w'Rw'\) by the assumption of Proposition 19. By Lemma 30 \(F'\) is connected, and thus \(F'\vDash \mathbf {DT_1CK}\). By Lemma 33 there is a reduced global path \(\alpha = a_1 \ldots a_m\) in \(F'\) such that \( a_1 = w'\). Let
be an infinite path shuttling back and forth through \(\alpha \). Rename the points in \(\upgamma \):
Again as in case (a3) we put \(F_j {:}= F | \overline{R} (b_j)\), and assume that \(c_j \in C_j\) and \(C_j\) is an \(R\)-maximal cluster. By IH there exist
and by Proposition 11 there exist \(f_{2j+1}: \mathbf{I } {\Delta }_{2j+1} \twoheadrightarrow ^d{}C_j\). Put
One can check that \(f\) is a d-morphism (Fig. 11.8).
(a5) \(W = \overline{R}(b)\), \(\lnot b R b\) and \(b R_D b\). Then \(R_D\) is universal, \(w'\ne b\). Put
and let \(X_1,X_2\) be two disjoint closed balls in \(X'\), \(X_3 := X'- X_1 - X_2\).
Let \(C\) be a maximal cluster in \(R(w')\), \(F' := F|R(w')\). Then there exist:
Put \(f:=f_1\cup f_2\cup f_3\cup f_4\) (Fig. 11.8). Then \(f(\partial X) = \left\{ w'\right\} \).
By Lemma 16 (b), \(f_1\cup f_2:X_1\cup X_2\twoheadrightarrow ^d F\), and hence \(f:X\twoheadrightarrow ^d F\) by Lemma 16 (a). \(f\) is manifold at \(b\), thus it is a dd-morphism by Lemma 18.
Now we prove (b). There are three cases.
(b1) \(w' = w'' = b\) and \(W = R(b)\). The argument is the same as in case (a1), using Proposition 11, Lemma 15, the induction hypothesis, and Proposition 12.
(b2) \(w' = w'' = b\), but \(W \ne R(b)\). Consider a maximal cluster \(C \subseteq R (b)\). Since all spherical shells for different \(r_1\) and \(r_2\) are homeomorphic, we assume that \(r_1 = 1\), \(r_2 = 4\). Consider the sets
and let \(X_0\subset X'\) be a closed ball, \(X_2 := X' - X_0\). Let \(F' :=F| R (b)\). There exist
One can check that \(f:X\twoheadrightarrow ^{dd}F\) for \(f:=f_0\cup f_1\cup f_2\cup f_3\).
(b3) \(w' \ne w''\) and for some \(b\in W\), \(W = R(b)\), so \(F\) has an \(R\)-reflexive root. Let
and let \(C_i\) be an \(R\)-maximal cluster in \(F_i\) for \(i\in \left\{ 1,2\right\} .\)
We assume that \(r_1 = 1\), \(r_2 = 6\) and consider the sets
By case (b1) and Proposition 11 we have
One can check that \(f:X\twoheadrightarrow ^{dd}F\) for \(f:= \bigcup \limits _{i=1}^5 f_i\) [Fig. 11.9, Case (b3)].
(b4) \(w' \ne w''\) and \(W \ne R(b)\) for any \(b\in W\). By Lemma 32 there is a reduced path \(\alpha = b_0 c_0 b_1 \ldots c_{m-1} b_m\) from \(b_0 = w'\) to \(b_m = w''\) that does not contain \(R_D\)-irreflexive points, \(c_i \in C_i\), where \(C_i\) is an \(R\)-maximal cluster. We may also assume that
Indeed, if the frame \((W,R)\) is not rooted, then (11.7) obviously holds. If \((W,R)\) is rooted, then its root \(r\) is irreflexive and by Lemma 30, \(R(r)\) is connected, so there exists a path \(\alpha \) in \(R(r)\) satisfying (11.7). Put
Assuming that \(r_1 = 1\), \(r_2 = 2m+1\) we define
By cases (b2), (b1), Proposition 11, and the induction hypothesis there exist
We claim that \(f:X\twoheadrightarrow ^{dd}F\) for \(f:= \bigcup \limits _{i=0}^{2m} f_i\) (Fig. 11.10). First, we prove by induction using Lemma 16 (see the previous cases) that \(f\) is a d-morphism. Note that \(f(Y')= f(Y_0) = \left\{ w'\right\} \) and \(f(Y'')= f(Y_{2m+1}) = \left\{ w''\right\} \).
Second, there are no \(R_D\)-irreflexive points in \(\alpha \), so all preimages of \(R_D\)-irreflexive points are in \(\Delta _0\); since \(f_0\) is a dd-morphism, \(f\) is 1-fold at any \(R_D\)-irreflexive point and manifold at all the others. Thus \(f\) is a dd-morphism by Proposition 12. \(\square \)
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Kudinov, A., Shehtman, V. (2014). Derivational Modal Logics with the Difference Modality. In: Bezhanishvili, G. (eds) Leo Esakia on Duality in Modal and Intuitionistic Logics. Outstanding Contributions to Logic, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8860-1_11
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