Generalized L-ˇCech closure operators and beyond

In this paper, we first give a concrete example to illustrate that the construction of L-ˇCech closure operators given in [24, Theorem 3.1] fails to hold when L is a continuous residuated lattice. By strengthening L to be completely distributive, we propose two new kinds of operators, which are respectively called generalized closure operators and generalized L-closure operators, and establish the categorical relation between them. As an application, we prove the existence of a Galois connection between the category of stratified generalized L-closure spaces and the category of stratified L-convex spaces. As a natural extension of probabilistic uniform spaces, we further introduce and study L-ˇCech uniform spaces, and also present their connection with L-ˇCech closure spaces.