an:05563897
Zbl 1172.03013
ten Cate, Balder; Gabelaia, David; Sustretov, Dmitry
Modal languages for topology: expressivity and definability
EN
Ann. Pure Appl. Logic 159, No. 1-2, 146-170 (2009).
0168-0072
2009
j
03B45 54A05 03C40
modal logic; topology; expressivity; definability
The topological language \(\mathcal{L}_t\) [\textit{J. Flum} and \textit{M. Ziegler}, Topological model theory. Lecture Notes in Mathematics. 769. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0421.03024)] has variables \(x,y,\ldots\) for points and \(U,V,\dots\) for open sets of a topological space. Predicate symbols \(P_p\) correspond to propositional variables \(p\). Quantifiers for sets are allowed only in combinations \(\forall U(x\in U\rightarrow\alpha)\) when \(\alpha\) is positive in \(U\). The authors prove two characterization theorems.
Theorem. A formula \(\phi(x)\) of \({\mathcal L}_t\) is equivalent to a standard translation of a propositional formula iff it is invariant under topo-bisimulations.
Theorem. A class \(K\) of topological spaces definable in \({\mathcal L}_t\) is definable in the basic modal language iff \(K\) is closed under topological sums, open subspaces and images of interior maps, while the complement of \(K\) is closed under Alexandroff extensions.
G. E. Mints (Stanford)
0421.03024