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

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 $\mathrm{\Sigma }P\times \mathrm{\Sigma }Q$ is a Fréchet space, then $\mathrm{\Sigma }(P\times Q)=\mathrm{\Sigma }P\times \mathrm{\Sigma }Q$. (2) For every $\omega$-well-filtered coherent $d$-space $X$, if $X\times X$ is a Fréchet space, then $X$ is sober; (3) For every $\omega$ type P-space or consonant Wilker space $X$, $\mathrm{\Sigma }\mathcal{O}(X)$ is sober.