Local compactness does not always imply spatiality
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913972759822336 |
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| author | Bezhanishvili, G. Melzer, S. D. Raviprakash, R. Suarez, A. L. |
| author_facet | Bezhanishvili, G. Melzer, S. D. Raviprakash, R. Suarez, A. L. |
| contents | It is a well-known result in pointfree topology that every locally compact frame is spatial. Whether this result extends to MT-algebras (McKinsey-Tarski algebras) was an open problem. We resolve it in the negative by constructing a locally compact sober MT-algebra which is not spatial. We also revisit Nöbeling's largely overlooked approach to pointfree topology from the 1950s. We show that his separation axioms are closely related to those in the theory of MT-algebras with the notable exception of Hausdorffness. We prove that Nöbeling's Spatiality Theorem implies the well-known Isbell Spatiality Theorem. We then generalize Nöbeling's Spatiality Theorem by proving that each locally compact $T_{1/2}$-algebra is spatial. The proof utilizes the fact that every nontrivial $T_{1/2}$-algebra contains a closed atom, which we show is equivalent to the axiom of choice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local compactness does not always imply spatiality Bezhanishvili, G. Melzer, S. D. Raviprakash, R. Suarez, A. L. General Topology 54D45, 54D30, 54D10, 06E25, 06D22, 18F70, 03E25 It is a well-known result in pointfree topology that every locally compact frame is spatial. Whether this result extends to MT-algebras (McKinsey-Tarski algebras) was an open problem. We resolve it in the negative by constructing a locally compact sober MT-algebra which is not spatial. We also revisit Nöbeling's largely overlooked approach to pointfree topology from the 1950s. We show that his separation axioms are closely related to those in the theory of MT-algebras with the notable exception of Hausdorffness. We prove that Nöbeling's Spatiality Theorem implies the well-known Isbell Spatiality Theorem. We then generalize Nöbeling's Spatiality Theorem by proving that each locally compact $T_{1/2}$-algebra is spatial. The proof utilizes the fact that every nontrivial $T_{1/2}$-algebra contains a closed atom, which we show is equivalent to the axiom of choice. |
| title | Local compactness does not always imply spatiality |
| topic | General Topology 54D45, 54D30, 54D10, 06E25, 06D22, 18F70, 03E25 |
| url | https://arxiv.org/abs/2508.01645 |