# The Canonical Lattice Isomorphism between Topologies Compatible with a Linear Space

## TL;DR

This paper establishes a canonical lattice isomorphism between compatible topologies on finite-dimensional vector spaces over valuation fields and subspace lattices, aiding in understanding their structure and continuous maps.

## Contribution

It constructs a canonical lattice isomorphism linking compatible topologies and subspace lattices, and characterizes continuous linear maps and Hausdorff topologies in this context.

## Key findings

- Canonical lattice isomorphism between topologies and subspaces
- Characterization of continuous linear maps
- Description of all Hausdorff compatible topologies

## Abstract

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.