z-Compact spaces

z-compact and z-Lindelof spaces are studied. Characterizations of z-compact spaces and z- Lindelof spaces, using multifunctions, are given.Our main results are .A space X is z-compact iff for every space Y and z-closed graph multifunction on X into Y the image of every z-closed set in X, is closed in Y. A space X is z-Lindelof iff for every P-space Y and z-closed graph multifunction on X into Y the image of every z-closed set in X is closed in Y.