  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 $\theta$-continuous, strongly and weakly
$\theta$-continuous, weakly and super-continuous.
