Fréchet spaces, ω-Rudin property and Smyth power spaces

For a $T_0$-space X, let $K\left(X\right)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order (i.e., the reverse inclusion order). The Smyth power poset $K\left(X\right)$ equipped with the upper Vietoris topology is called the Smyth power space of X and is denoted by $P_S\left(X\right)$. This paper is mainly devoted to discuss some properties of Fréchet spaces from the viewpoint of intersection of topology and domain theory. We prove that if the sobrification (especially, the Hoare power space) of a $T_0$-space X is a Fréchet space, then X is an ω-Rudin space. Hence every ω-well-filtered space for which its sobrification (especially, its Hoare power space) is a Fréchet space is sober, and every second-countable ω-well-filtered space is sober. We also show that if the Smyth power space of a $T_0$-space X is a Fréchet space, then the Scott topology is coarser than the upper Vietoris topology on $K\left(X\right)$, whence if X is additionally well-filtered, then the topology of Smyth power space $P_S\left(X\right)$ is the Scott topology of $K\left(X\right)$. Moreover, if X is a second-countable ω-well-filtered space, then the topology of $P_S\left(X\right)$ is the Scott topology of $K\left(X\right)$.