Abstract. In previous papers, various notions of compact, T3, T4, and Tychonoff objects in a topological category were introduced and compared. The main objective of this paper is to characterize each of these classes of objects in the categories of filter and local filter convergence spaces as well as to examine how these various generalizations are related.

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.

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In this paper, the characterization of closed and strongly closed subobjects of an object in category of semiuniform convergence spaces is given and it is shown that they induce a notion of closure which enjoy the basic properties like idempotency,(weak) hereditariness, and productivity in the category of semiuniform convergence spaces. Furthermore, T1 semiuniform convergence spaces with respect to these two new closure operators are characterized.