Abstract
We develop a theory of compactness in the category of fuzzy convergence spaces in the spirit of Lowen, i.e. we define for each fuzzy subset of a fuzzy convergence space a degree of being compact. These degrees of compactness generalize at the same time the theory of compactness in the category of fuzzy convergence spaces and some known measures of compactness in fuzzy topological spaces. Together with degrees of closedness and of even continuity they prove useful in the development of a theory of compactness for fuzzy function spaces.
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