Abstract
Within ZFC, we develop a general technique to topologize trees that provides a uniform approach to topological completeness results in modal logic with respect to zero-dimensional Hausdorff spaces. Embeddings of these spaces into well-known extremally disconnected spaces then gives new completeness results for logics extending S4.2.
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Notes
The notion of densely discrete has appeared in the literature under the names \(\alpha \)-scattered [26] and weakly scattered (see, e.g., [6, 7, 11]). Since the term weakly scattered is often used for a different concept (see, e.g., [24, p. 120]), we adopt Arhangel’skii’s more descriptive terminology.
This can be seen as follows. Theorem 6.10(2) implies that there is a basis for \(\mathbb T_{\omega _1}^\omega \) consisting of \(\omega _1\) clopen subsets. Therefore, since \(\mathbb T_{\omega _1}^\omega \) is Lindelöf, it has at most continuum many clopen sets. Thus, the weight of \(\beta (\mathbb T_{\omega _1}^\omega )\) is at most \(\mathfrak c\), and hence \(\beta (\mathbb T_{\omega _1}^\omega )\) embeds into the Cantor cube \(2^\mathfrak c\). By [17, Sec. 2], \(\beta (\mathbb T_{\omega _1}^\omega )\) embeds into the Gleason cover of \(2^\mathfrak c\), which by Efimov’s theorem (see, e.g., [17]) embeds into \(\beta (\omega )\).
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Acknowledgements
We would like to thank the referee for careful reading, useful comments, and suggesting a simplification of the proof of Theorem 5.9.
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Bezhanishvili, G., Bezhanishvili, N., Lucero-Bryan, J. et al. Tree-like constructions in topology and modal logic. Arch. Math. Logic 60, 265–299 (2021). https://doi.org/10.1007/s00153-020-00743-6
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DOI: https://doi.org/10.1007/s00153-020-00743-6
Keywords
- Modal logic
- Topological semantics
- Tree
- Tychonoff space
- Zero-dimensional space
- Scattered space
- Extremally disconnected space
- P-space
- Lindelöf space
- Patch topology