On Function Spaces Related to H-sober Spaces

Abstract

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 $f:X⟶Y$ 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.

11 pages