Xi-Zhao Model Preserves WD Spaces

Abstract

For a $T_1$ space $X$, Zhao and Xi constructed a dcpo model $\hat{P}$, where $P$ is a bounded complete algebraic poset model of $X$. In this paper, we formulate the closed WD subsets of the maximal point space $\mathrm{Max}(\hat{P})$ and the Scott space $\Sigma \hat{P}$, and then prove that $X$ is a WD space if and only if $\Sigma \hat{P}$ is a WD space. It is also shown that the sobrification $X^s$ (resp., the well-filtered reflection $X^w$) of $X$ can be embedded into the sobrification ${\hat{P}}^s$ (resp., the well-filtered reflection ${\hat{P}}^w$) of $\hat{P}$ as a saturated subspace. Finally, we introduce two new concepts ``$\mathrm{H}$-model spaces" and ``weak $\mathrm{H}$-model spaces", and provide a general framework to prove that such $T_1$ spaces can be preserved by Xi-Zhao dcpo models.