T4 , Urysohn’s lemma, and Tietze extension theorem for constant filter convergence spaces

In 1978, Schwarz [14] introduced the category ConF CO whose objects are constant filter convergence spaces and morphisms are continuous maps, and he showed that ConF CO is isomorphic to the category FILTER whose objects are filter spaces and morphisms are continuous maps. He also showed that it is a bireflective subcategory of F CO whose objects are filter convergence spaces and morphisms are continuous maps. Hence, Schwarz proved that ConF CO is the natural link between FILTER and the category F CO. In 1991, Baran [3] introduced the local T1 separation property that is used to define the notion of strongly closed subobject of an object of a topological category, which are used in the notions of compactness [8], connectedness [10], and normal objects [3]. In general topology, one of the most important uses of separation properties is theorems such as the Urysohn?s lemma and the Tietze extension theorem. In this regard, it is useful to be able to extend these In this paper, we characterize various local forms of T4 constant filter convergence spaces and investigate the relationships among them as well as showing that the full subcategories of the category of constant filter convergence spaces consisting of local T4 constant filter convergence spaces that are hereditary. Furthermore, we examine the relationship between local T4 and general T4 constant filter convergence spaces. Finally, we present Urysohn?s lemma and Tietze extension theorem for constant filter convergence spaces.