Let be a family of subsets of a topological space X. We do not require to be a covering of X, nor do we assume that the members of are necessarily open. In this paper we shall assume that is of a special sort, which we call Σ-Finite. We show that a Σ-Finite family is both locally finite and star-finite, and in particular that an open covering of X is Σ-Finite if and only if it is star-finite. We then prove that every Σ-Finite family is ᓂ-discrete, so that in particular, every star-finite open covering of X is (ᓂ-discrete. There seems to be some applications of this fact.