UPPER AND LOWER QUASI cl-SUPERCONTINUOUS MULTIFUNCTIONS
J. K. KOHLI 1, C. P. ARYA 2 AND D. SINGH 3 1. Department of Mathematics, Hindu College, University of Delhi, Delhi-110007, INDIA, e-mail: jk kohli@yahoo.com
2. ISSA/DRDO, Metcalfe House, Delhi 110054, INDIA, e-mail: car-ya28@gmail.com
3. Department of Mathematics, Sri Aurobindo College, University of Delhi, New Delhi 110017, INDIA, e-mail:dstopology@rediffmail.com
The notion of quasi cl-supercontinuity of functions is extended to the framework of multifunctions. Basic properties of upper (lower) quasi cl-supercontinuous multifunctions are studied and their place in the hierarchy of variants of continuity of multifunctions, that already exist in the literature, is elaborated. The class of upper (lower) quasi cl-supercontinuous multifunctions properly contains the class of upper (lower) cl-supercontinuous multifunctions and so includes all upper (lower) perfectly continuous multifunctions; and is strictly contained in the class of upper (lower) quasi z-supercontinuous multifunctions. The upper quasi cl-supercontinuity of multifunctions is preserved under compositions, union of multifunctions, restriction to a subspace and the passage to the graph multifunction. A sufficient condition for the intersection of two upper quasi cl-supercontinuous multifunctions to be upper quasi cl-supercontinuous is formulated. The lower quasi cl-supercontinuity of multifunctions is preserved under the shrinking and expansion of range, union of multifunctions and restriction to a sub-space.
