On function spaces related to some kinds of weakly sober spaces

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.