Quotient reflective subcategories of the category of bounded uniform filter spaces

Abstract

Previously, several notions of $ T_{0} $ and $ T_{1} $ objects have been studied and examined in various topological categories. In this paper, we characterize each of $ T_{0} $ and $ T_{1} $ objects in the categories of several types of bounded uniform filter spaces and examine their mutual relations, and compare that with the usual ones. Moreover, it is shown that under $ T_{0} $ (resp. $ T_{1} $) condition, the category of preuniform (resp. semiuniform) convergence spaces and the category of bornological (resp. symmetric) bounded uniform filter spaces are isomorphic. Finally, it is proved that the category of each of $ T_{0} $ (resp. $ T_{1} $) bounded uniform filter space are quotient reflective subcategories of the category of bounded uniform filter spaces.

Keywords

• topological category
• bounded uniform filter space
• t₀ objects
• t₁ objects
• initial and final lifts
• quotient-reflective