A set A in a topological space X is called kappa-closed if (B) over bar subset of A whenever B subset of A and \B\ < kappa. A kappa-hole in X is a maximal centered family of kappa-closed sets which is both kappa-complete and free. A family S of finite subsets of X is called kappa-closed if u boolean OR {x: u boolean OR {x} is an element of S} is kappa-closed in X for every u is an element of [X]( omega, X is an initially < kappa-compact T-1 space and U is an open cover of X such that for every A is an element of [X](kappa) there is a set B is an element of [A](

Minimal covering of infinite sets is studied. It is shown that if M subset-of C2x is closed, then M contains a minimal covering of X. If M is a covering of X, then M is closed on X if and only if there is an admissible function f: X-->M which has no chain.For star type set systems Theorem 15 gives a sufficient condition to contain finite covering.