Summary: In this article we prove a global result in the spirit of Basener’s theorem regarding the relation between \(q\)-pseudoconvexity and \(q\)-holomorphic convexity: we prove that any open subset \(\Omega \subset \mathbb{C}^n\) with smooth boundary, strictly \(q\)-pseudoconvex, is \(( q + 1 )\)-holomorphically convex; moreover, assuming that \(\Omega\) verifies an additional assumption, we prove that it is \(q\)-holomorphically convex. We also prove that any open subset of \(\mathbb{C}^n\) is \(n\)-holomorphically convex.