PreT 2 Objects in Topological Categories

In previous papers, two notions of pre-Hausdorff (PreT (2)) objects in a topological category were introduced and compared. The main objective of this paper is to show that the full subcategory of PreT (2) objects is a topological category and all of T (0), T (1), and T (2) objects in this topological category are equivalent. Furthermore, the characterizations of pre-Hausdorff objects in the categories of filter convergence spaces, (constant) local filter convergence spaces, and (constant) stack convergence spaces are given and as a consequence, it is shown that these categories are homotopically trivial.