Zero-Dimensionality and Hausdorffness in Quantale-Valued Preordered Spaces

Abstract

In this paper, we study the category of quantale-valued preordered spaces. We show that it is a normalized topological category and give characterization of zero-dimensionality and $D$-connectedness in the category of quantale-valued preordered spaces. Moreover, we characterize explicitly each of $\overline{T_{0}}$, $T_{0}$, $T_{1}$, pre-$\overline{T_{2}}$, $\overline{T_{2}}$ and $NT_{2}$ quantale-valued preordered spaces. Finally, we examine how these characterization are related to each other and show that the full subcategory $\bf{T}_{i}$$(\bf{pre}$-$\bf{T_{2}} (\mathcal{L}\textbf{-Prord}))$ ($i=0,1,2$) of $\bf{pre}$-$\bf{T_{2}} (\mathcal{L}\textbf{-Prord})$, and the full subcategory $\bf{T}_{i}$ $(\mathcal{L}\textbf{-Prord})$ ($i=1,2$) of $\mathcal{L}\textbf{-Prord}$ are isomorphic.