arXiv subjects

Takanobu Aoyama

Publications and source records attributed to Takanobu Aoyama.

2 recordsLinked to original sources

arXiv2023

On Rigidity of Lattice of Topologies Containing Vector 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 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.

math.GN↗

arXiv2020

The Canonical Lattice Isomorphism between Topologies Compatible with a Linear Space

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.

math.GN↗