The answer to your question is yes, but it is a stretch to claim that the topology is due to Furstenberg. There is an extended discussion on Furstenberg's proof in the comments to this answer. The short version is as Chandan Singh Dalawat said in the comments above: this topology on the integers is the profinite topology, and people had been studying profinite topologies long before Furstenberg.
The topology is useful in the sense that profinite completions are useful. In particular, you may argue that it is a natural topology on the fundamental group of a circle (or the punctured complex affine line), since its profinite completion is the geometric fundamental group of the multiplicative group $\mathbb{G}_m$. It also appears in some form whenever one uses the ring of adeles $\mathbb{A}_\mathbb{Q}$, which you may encounter when studying Tate's thesis or automorphic representations.
