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 T0, T0, T1, pre-T2, T2 and NT2 quantale-valued preordered spaces. Finally, we examine how these characterization are related to each other and show that the full subcategory Ti(pre-T2(L-Prord)) (i=0,1,2) of pre-T2(L-Prord), and the full subcategory Ti (L-Prord) (i=1,2) of L-Prord are isomorphic.