Modal languages for topology: expressivity and definability (Q1023054)

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.