Closure Operators and Connectedness in Bounded Uniform Filter Spaces

Abstract

In this paper, we characterize both closed and strongly closed subobjects in the category of bounded uniform filter spaces and introduce two notions of closure operators which satisfy weakly hereditary, idempotent and productive properties. We further characterize each of $T_j$ ($j = 0, 1$) bounded uniform filter spaces using these closure operators and examine that each of them form quotient-reflective subcategories of the category of bounded uniform filter spaces. Also, we characterize connected bounded uniform filter spaces. Finally, we introduce ultraconnected objects in topological category and examine the relationship among irreducible, ultraconnected and connected bounded uniform filter spaces.