On the Minimal Covering of Infinite Sets

Klimb, J., On the minimal covering of infinite sets, Discrete Applied Mathematics 45 (1993) 161-168. Minimal covering of infinite sets is studied. It is shown that if BC2’ is closed, then /* contains a minimal covering of X. If r// is a covering of X, then fl is closed on X if and only if there is an admissible functionf: X-1 4 which has no chain. For star type set systems Theorem 15 gives a sufficient condition to contain finite covering. Notations. Let X be a set. We denote by 2x the set of all subsets of X. If B c A and B#A, we write BcA. For J&C~~, Y~Xlet &II Y={Mfl Y: MEA}. Definition 1. The set A is a covering of the set X if (1) &~2~ and (2) U&d M=X. Definition 2. JX is a minimal covering of X if (1) A is a covering of X and (2) A!'c&! implies that A’ is not a covering of X. Definition 3. Let .A be a covering of X. A is closed on X if there is a well-ordering P of & with the following property: for all x E X, the set {ME A: x E M} has a maximal element according to the well-ordering ,D. Remarks. (1) If A? is a point finite covering of X (i.e., VXE X 1 {ME .,tv: x E Correspondence to: Dr. J. KlimO, Research Institute for Telecommunication, POB 15, I525 Budapest, Hungary. 0166-218X/93/$06.00