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.

In this paper, we mainly study function spaces related to some kinds of weakly sober spaces, such as bounded sober spaces, $k$-bounded sober spaces and weakly sober spaces. For $T_0$ spaces $X$ and $Y$, it is proved that $Y$ is bounded sober iff the function space $\mathbf{T}\mathbf{o}\mathbf{p}(X,Y)$ of all continuous functions $f:X⟶Y$ equipped with the pointwise convergence topology is bounded sober iff $\mathbf{T}\mathbf{o}\mathbf{p}(X,Y)$ equipped with the Isbell topology is bounded sober. But for a $k$-bounded sober space $X$, the function space $\mathbf{T}\mathbf{o}\mathbf{p}(X,Y)$ equipped with the pointwise convergence topology or the Isbell topology may not be $k$-bounded sober. It is shown that if the function space $\mathbf{T}\mathbf{o}\mathbf{p}(X,Y)$ equipped with the pointwise convergence topology or the Isbell topology is weakly sober (resp., a cut space), then $Y$ is weakly sober (resp., a cut space). Relationships among some kinds of (weakly) sober spaces are also investigated.

In ^[16], using Rudin sets, Miao, Li and Zhao introduced a new concept of weakly well-filtered spaces— $k$-bounded well-filtered spaces. Now, also using Rudin sets, we introduce another type of $T_0$ spaces—weakly bounded well-filtered spaces, which are strictly stronger than $k$-bounded well-filtered spaces. Some basic properties of $k$-bounded well-filtered spaces and weakly bounded well-filtered spaces are investigated and the relationships among some kinds of weakly sober spaces and weakly well-filtered spaces are posed. It is proved that the category $\mathbf{K}\mathbf{B}\mathbf{W}\mathbf{F}$ is not reflective in the category ${\mathbf{T}\mathbf{o}\mathbf{p}}_0$.