On the Rigidity of Lattices of Topologies on Vector Spaces

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.