T 2 -objects in topological categories

Abstract

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.

T3 AND T4-objects in the topological category of cauchy spaces

“…(14) X is N T 2 i¤ X is T 0 and P reT 2 [12]. (ii) By Theorem 3.1(1) of [6], if X is P reT 0 2 , then X is P reT 2 .…”

“…We now recall, ( [2], [7] and [13]), various generalizations of the usual T 3 separation axiom to arbitrary set based topological categories and characterize each of them for the topological categories CHY. [7] and [13]) Let U : E !…”

“…We now recall various generalizations of the usual T 4 separation axiom to arbitrary set based topological categories that are de…ned in [2], [7] and [13], and characterize each of them for the topological categories CHY.…”

“…Moreover, various generalizations of each of T i , i = 0; 1; 2 separation properties for an arbitrary topological category over SET, the category of sets are given and the relationship among various forms of each of these notions are investigated by Baran in [2], [7], [8], [10], [12] and [14].…”

“…Recall, also, in [2], that there are various ways of generalizing each of the usual T 0 and T 2 separation axioms to topological categories. Moreover, the relationships among various forms of T 0 -objects and T 2 -objects are established in [11] and [12], respectively.…”