A pre-topological space equipped with an order is called an ordered pretopological space. These spaces form the objects of a category which will be denoted by OVPT. The arrows of this category are certain increasing maps called V -continuous. Essentially, we will prove that the subcategory of ordered pretopological spaces of type T 0 , OVPT 0 , is reflective in OVPT. We introduce and study some new separation axioms and characterize the class of morphisms orthogonal to the objects of OVPT 0 .

We give the T 0-reflection in the category of pretopological spaces with p-continuous maps as arrows. After that we will study some separation axioms in this category.