When do weakly first-countable spaces and the Scott topology of open set lattice become sober?

Abstract

In this paper, we investigate the sobriety of weakly first-countable spaces and give some sufficient conditions that the Scott topologies of the open set lattices are sober. The main results are: (1) Let $P$ and $Q$ be two posets. If $\Sigma P\times \Sigma Q$ is a Fr\'{e}chet space, then $\Sigma (P\times Q)=\Sigma P \times \Sigma Q$. (2) For every $\omega$-well-filtered coherent $d$-space $X$, if $X\times X$ is a Fr\'{e}chet space, then $X$ is sober; (3) For every $\omega$ type P-space or consonant Wilker space $X$, $\Sigma\mathcal{O}(X)$ is sober.