A NOTE ON CONTINUOUS-MAPPINGS AND THE PROPERTY OF KELLEY,J.L.

In this paper, it is proved that if X is a continuum and omega is any Whitney map for C(X), then the following are equivalent: (1) X has property [K]. (2) There exists a (continuous) mapping F: X x I x [0, omega(X)] --> C(X) such that F({x} x I x {t}) = {A is-an-element-of omega-1 (t)\x is-an-element-of A} for each x is-an-element-of X and t is-an-element-of [0, omega(X)], where I = [0, 1]. (3) For each t is-an-element-of [0, omega(X)], there is an onto map f: X x I --> omega-1 (t) such that f({x} x I) = {A is-an-element-of omega-1 (t)\x is-an-element-of A} for each x is-an-element-of X. Some corollaries are obtained also.