Compactness with Respect to a Convergence Structure

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A convergence structure for a category is given by determining convergent nets and their limits for each object of this category under validity of some basic convergence axioms. The nets considered are obtained as a categorical generalization of the usual nets. A convergence structure for a category induces, under some natural conditions, a closure operator of this category. We study separatedness and compactness of objects of a given category with respect to a convergence structure and show that they behave more naturally than the usual separatedness and compactness of topological spaces.