On three problems about well-filteredness of $T_0$-spaces

In this paper, we show that there is a countable Noetherian complete lattice $L$ and an order-compatible $d$-topology $\tau$ on $L$ such that $(L, \tau)$ is not well-filtered, and there exist a dcpo $P$ and an order-compatible well-filtered topology $\tau$ on $P$ but the Scott topology $\sigma (P)$ is not well-filtered.

For such poset $P$ and topology $\tau$, let $Y=(P, \tau)$ and $X = 1$ (the topological space with single point), then the function space $\mathbb{C}(X, Y)$ equipped with the Scott topology is not well-filtered.

These results answer three open problems concerning the well-filteredness of $T_0$-spaces.