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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
##### Keywords:
modal logic; topology; expressivity; definability
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