A generalized topological Rudin lemma and sober spaces

In this paper, we introduce and study the notion of $F$-families in $T_0$-spaces. This concept unifies several important classes of objects in domain theory and non-Hausdorff topology, including irreducible sets, filtered families of upper sets, Scott open filters, and irreducible subsets in Smyth power spaces. Using $F$-families, we establish a generalized topological Rudin lemma that extends the classical topological Rudin lemma to a broader setting. As an application, we obtain some characterizations of sober spaces, from which the Hofmann-Mislove theorem and Heckman-Keimel-Schalk theorem can be directly deduced.