  We consider all compatible topologies of an arbitrary finite-dimensional
vector space over a non-trivial valuation field whose metric completion is a
locally compact space. We construct the canonical lattice isomorphism between
the lattice of all compatible topologies on the vector space and the lattice of
all subspaces of the vector space whose coefficient field is extended to the
complete valuation field. Moreover, in this situation, we use this isomorphism
to characterize the continuity of linear maps between finite-dimensional vector
spaces endowed with given compatible topologies, and also, we characterize all
Hausdorff compatible topologies.
