Proper and admissible topologies in the setting of closure spaces

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A Cech closure space (X,u) is a set X with a (Cech) closure operator u which need not be idempotent. Many properties which hold in topological spaces hold in Cech closure spaces as well. The notions of proper (splitting) and admissible (jointly continuous) topologies are introduced on the sets of continuous functions between Cech closure spaces. It is shown that some well-known results of Arens and Dugundji and of Iliadis and Papadopoulos are true in this setting. We emphasize that Theorems 1--10 encompass the results of A. di Concilio and of Georgiou and Papadopoulos for the spaces of continuous-like functions as θ-continuous, strongly and weakly θ-continuous, weakly and super-continuous.