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We introduce and study a concept of neighborhoods with respect to a categorical closure operator. The concept, which is based on using pseudocomplements in subobject lattices, naturally generalizes the classical neighborhoods in topological spaces and we show that it behaves accordingly. We investigate also separation and compactness dened in a natural way by the help of the neighborhoods introduced.

We introduce and study a concept of a convergence structure on a concrete category. The concept is based on using certain generalized filters for expressing the convergence. Some basic properties of the convergence structures are discussed. In particular, we study convergence separation and convergence compactness and investigate relationships between the convergence structures and the usual closure operators on categories.Keywords Concrete category · Subobject lattice · Filter · Raster · Convergence structure on a category · Closure operator on a category · Separation · Compactness Mathematics Subject Classifications (2000) Primary 18D35 · 54A20 · Secondary 54A05 · 54B30 · 54A20