On weakly bounded well-filtered spaces

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 ${\mathbf{T}\mathbf{o}\mathbf{p}}_0$.