F-Net Convergence with Respect to a Closure Operator

Abstract

In a category with a closure operator, we introduce the notion of a neighborhood of a point. The neighborhoods are then discussed and used for defining a net convergence structure on the category with respect to the closure operator. The nets considered are obtained as a categorical generalization of the usual nets. We investigate basic properties of the defined convergence. In particular, we study convergence separation and convergence compactness and describe their relationships to the usual closure separation and closure compactness.