  In this paper, the concept of quasi-finitely separating map and
quasiapproximate identity are introduced. Based on these concepts, QFS-spaces
and quasicontinuous maps are defined. Properties and characterizations of
QFS-spaces are explored. Main results are: (1) Each QFS-space is
quasicontinuous space; (2) Closed subspaces, quasicontinuous projection spaces
of QFS-spaces are QFS-spaces; (3) Continuous retracts of QFS-spaces are
QFS-spaces and a kind of retracts of QFS-spaces are constructed; (4) Upper
powerspaces of continuous QFS-spaces are FS-spaces.
