Suppose that f_{i}:A_{i －}> B_{i} is an upper hemicontinuous correspondence for 1 i n.
Define F:A_{1}A_{2}...A_{n} －> B_{1}B_{2}...B_{n} to be a correspondence such that, for all a=(a_{1},a_{2},...,a_{n}) belonging to A_{1}A_{2}...A_{n}, F(a):= f_{1}(a_{1})f_{2}(a_{2})...f_{n}(a_{n}).
How do I show that F is also upper hemicontinuous?
Here,
definition 1:
A correspondence F: X－> Y is a map F from X to "subsets of Y".
definition 2:
A correspondence F: X－> Y is upper hemicontinuous if, for each sequence {x_{k}} X converging to x' belonging to X and each open set Y* Y such that F(x') Y*, there is a natural number k_{0} such that, for each k k_{0}, F(x_{k}) Y*.