In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{ 0}$ spaces $X$ and $Y$, it is proved that $Y$ is H-sober iff the function space $\mathbb{ C}(X, Y)$ of all continuous functions $f : X\longrightarrow Y$ equipped with the topology of pointwise convergence is H-sober iff the function space $\mathbb{ C}(X, Y)$ equipped with the Isbell topology is H-sober. One immediate corollary is that for a $T_{ 0}$ space $X$, $Y$ is a sober space (resp., $d$-space, well-filtered space) iff the function space $\mathbb{ C}(X, Y)$ equipped with the topology of pointwise convergence is a sober space (resp., $d$-space, well-filtered space) iff the function space $\mathbb{ C}(X, Y)$ equipped with the the Isbell topology is a sober space (resp., $d$-space, well-filtered space). It is shown that $T_{ 0}$ spaces $X$ and $Y$, if the function space $\mathbb{ C}(X, Y)$ equipped with the compact-open topology is H-sober, then $Y$ is H-sober. The function space $\mathbb{ C}(X, Y)$ equipped with the Scott topology is also discussed.
Categories: math.GN