On lattice isomorphisms between lattice of topologies on linear spaces (New developments of transformation groups)

Abstract

We consider a lattice isomorphism between the lattice of topologies on linear spaces whose dimensions are at least two over Hausdorff topological fields. By using a result of J. Hartmanis (1958), we show that if the image of the set of linear topologies on the domain space by the lattice isomorphism coincides with the set of linear topologies on the target space, then the coefficient fields are isomorphic and the linear spaces are semi-linear isomorphic. Moreover, such a lattice isomorphism or a composition of a involution with the lattice isomorphism is induced by a composition of a linear translation with the semi-linear isomorphism.