Topologie sur l'ensemble des parties positives d'un r\'eseau

TL;DR

This paper introduces a topology on the set of positive parts of a lattice, compares it with a related topology on a dual vector space, and studies its properties.

Contribution

It defines a new topological structure on positive parts of lattices and analyzes its properties in relation to dual vector space topologies.

Findings

The topology on positive parts is well-defined and meaningful.

Comparison with the topology of the dual vector space reveals structural insights.

Properties of the topology are characterized and understood.

Abstract

We define a notion of {\it positive part} of a lattice $\Lambda$ and we endow the set of such positive parts with a topology. We then study some properties of this topology, by comparing it with the one of $V^*/\RM_{> 0}$, where $V^*$ is the dual vector space of $\RM \otimes_\ZM \Lambda$.