  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.

General Topology