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

We define a notion of {\it positive part} of a lattice $\mathrm{\Lambda }$ 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 $V^{\ast }\mathrm{/}{\text{RM}}_{>0}$, where $V^{\ast }$ is the dual vector space of $\text{RM}{\otimes }_{\text{ZM}}\mathrm{\Lambda }$.