Abstract
A vector topology on a vector space over a topological field is a (not necessarily Hausdorff) topology by which the addition and the scalar multiplication are continuous. We prove that, if an isomorphism between the lattices of topologies of two vector spaces preserves vector topologies, then the isomorphism is induced by a translation, a semilinear isomorphism and the complement map. As a consequence, if such an isomorphism exists, the coefficient fields are isomorphic as topological fields and these vector spaces have the same dimension. We also prove a similar rigidity result for an isomorphism between the lattice of vector topologies which preserves Hausdorff vector topologies. To obtain these results, we construct a Galois connection between a lattice of vector topologies and a lattice of subspaces and use the fundamental theorems of affine and projective geometries.
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References
Aoyama, T.: The canonical lattice isomorphism between topologies compatible with a linear space and subspaces. Tsukuba J. Math. 47(1), 41–64 (2023)
Bael, R.: Linear algebra and projective geometry. Dover Publications, New York (2005)
Birkhoff, G.: On the combination of topologies. Fund. Math. 26, 156–166 (1936)
Bourbaki, N.: Topological vector spaces, 1981. (French) (English translated version: topological vector spaces Chapter 1–5, Translated by Eggleton, H.G., Madan, S., Springer-Verlag.) (1987)
Ellis, D.: On the topolattice and permutation group of infinite set. II, Proc. Cambridge Philos. Soc. 50, 485–487 (1954)
Hartmanis, J.: On the lattice of topologies. Canadian J. Math. 10, 547–553 (1958)
Scherk, P.: On the fundamental theorem of affine geometry. Canad. Math. Bull. 5, 67–69 (1962)
Steiner, E.F.: On finite dimensional linear topological spaces. Amer. Math. Monthly 72, 34–35 (1965)
Acknowledgements
The author would like to express his gratitude to Prof. Ken’ichi Ohshika and Prof. Shinpei Baba for valuable comments. He is grateful to the anonymous referee for reviewing the manuscript and suggesting improvements. He also would like to thank the Research Institute for Mathematical Sciences (RIMS) for giving him opportunities of research presentations on results in this paper at Kyoto University.
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Takanobu Aoyama conducted the research and wrote the whole manuscript text. He reviewed the manuscript.
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Aoyama, T. On the Rigidity of Lattices of Topologies on Vector Spaces. Order (2023). https://doi.org/10.1007/s11083-023-09655-5
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DOI: https://doi.org/10.1007/s11083-023-09655-5
Keywords
- Lattice of topologies
- Topological vector space
- Lattice isomorphism
- The fundamental theorem of affine geometry
- The fundamental theorem of projective geometry