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Modal languages for topology: expressivity and definability. (English) Zbl 1172.03013
The topological language \(\mathcal{L}_t\) [J. Flum and 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.

MSC:
03B45 Modal logic (including the logic of norms)
54A05 Topological spaces and generalizations (closure spaces, etc.)
03C40 Interpolation, preservation, definability
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