From his first works, Fréchet envisages, in a search for generality, the passage from functions defined on the real line to functions defined on abstract spaces. In order to guarantee the generalization of well-known theorems to the real case, Fréchet proposes various (pre-)topological notions and in particular a definition of compactness which is his own. Our philosophical approach is to show how Fréchet's notion of compactness is a particular way of working with and mastering an infinity linked to his notion of abstract space. We will show that this relation to infinity is different from the one we hear nowadays when we talk about compactness in elementary topology.