Filters and compactness on small categories and locales

In analogy with the classical theory of filters, for finitely complete or small categories, we provide concepts filter, \(\mathfrak{G}\)-neighborhood (short "Grothendieck-neighborhood") and cover-neighborhood points such to study convergence, cluster point, closure sieves compactness on objects that kind categories. Finally, all these in category \(\mathbf{Loc}\) locales.