  A vector topology on a vector space over a topological field is a (not
necessarily Hausdorff) topology by which the addition and scalar multiplication
are continuous. We prove that, if an isomorphism between the lattice 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. These results are obtained by using the fundamental theorems
of affine and projective geometries.
