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

Abstract

In this paper, we show that there is a countable Noetherian complete lattice $L$ and an order-compatible $d$-topology $τ$ on $L$ such that $(L, τ)$ is not well-filtered, and there exist a dcpo $P$ and an order-compatible well-filtered topology $τ$ on $P$ but the Scott topology $σ(P)$ is not well-filtered. For such poset $P$ and topology $τ$, let $Y=(P, τ)$ 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.