The McKinsey-Tarski theorem for locally compact ordered spaces. (English) Zbl 1483.54016
The modal logic \(\mathsf{L}(X)\) of a topological space \(X\) is the set of valid modal formulas, where box is interior and diamond is closure. The McKinsey-Tarski theorem says that the modal logic of any metrizable space without isolated points is \(\mathsf{S}4\). Various analogs of the theorem are known. The main result of the paper, Theorem 3.12, extends the McKinsey-Tarski Theorem to any nonempty locally compact space without isolated points which is a GO-space, that is, a subspace of a topological space coming from a linear order. It is open whether local compactness can be removed. Then Theorem 5.10 computes \(\mathsf{L}(X)\) for each nonempty locally compact GO-space.
Reviewer: Giacomo Lenzi (Fisciano)
MSC:
| 54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |
| 54D45 | Local compactness, \(\sigma\)-compactness |
| 03B45 | Modal logic (including the logic of norms) |
| 54G12 | Scattered spaces |
| 03B55 | Intermediate logics |
Keywords:
modal logic; topological semantics; linearly ordered space; generalized ordered space; locally compact space; scattered spaceReferences:
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