关键词: sober space, descriptive set theory, function space, strong $d$-space, product space, reflection

Abstract: This paper investigates sober spaces and their related structures from different perspectives. First, we extend the descriptive set theory of second countable sober spaces to first countable sober spaces. We prove that a first countable $T_{0}$ space is sober if and only if it does not contain a $\mathbf{\Pi}_{2}^{0}$-subspace homeomorphic either to $S_{D}$, the natural number set equipped with the Scott topology, or to $S_{1}$, the natural number set equipped with the co-finite topology, and it does not contain any directed closed subset without maximal elements either. Second, we show that if $Y$ is sober, the function space $TOP(X,Y)$ equipped with the Isbell topology (respectively, Scott topology) may be a non-sober space. Furthermore, we provide a uniform construction to $d$-spaces and well-filtered spaces via irreducible subset systems introduced in [9]; we called this an $\mathrm{H}$-well-filtered space. We obtain that, for a $T_{0}$ space $X$ and an $\mathrm{H}$-well-filtered space $Y$, the function space $TOP(X,Y)$ equipped with the Isbell topology is $\mathrm{H}$-well-filtered. Going beyond the aforementioned work, we solve several open problems concerning strong $d$-spaces posed by Xu and Zhao in [11].

Key words: sober space, descriptive set theory, function space, strong $d$-space, product space, reflection