Various types of upper semcontinuity properties for set-valued functions have been used in the past to obtain closure and lower closure theorems in optimal control theory as well as selection theorems and fixed-point theorems in topology. This paper unifies these various concepts by using semiclosure operators, extended topologies, and lattice theoretic operations and obtains general closure theorems. In addition, analytic criteria are given for this generalized upper semicontinuity. In particular, set-valued functions which are maximal in terms of certain properties (e.g., maximal monotone multifunctions) are shown to be necessarily upper semicontinuous.