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In this paper, we examine the category of ordered-RELspaces. We show that it is a normalized and geometric topological category and give the characterization of local T¯0, local T0′, and local T1 ordered-RELspaces. Furthermore, we characterize explicitly several notions of T0’s and T1 objects in O-REL and study their mutual relationship. Finally, it is shown that the category of T0’s (resp. T1) ordered-RELspaces are quotient reflective subcategories of O-REL.