A topological lattice on the set of multifunctions (Q1823501)

Summary: A Wilker space X is a topological space satisfying the following condition: For every compact subset \(K\subset Y\) and any pair of open subsets \(A_ 1,A_ 2\) of X with \(K\subset A_ 1\cap A_ 2\) there are compact subsets \(K_ 1\), \(K_ 2\) of X such that \(K\subset K_ 1\cap K_ 2\). Let M(X,Y) be the set of continuous multifunctions from a Wilker space X to a topological space Y equipped with the compact-open topology. M(X,Y) equipped with the partial order \(\subset\) is a topological \(\bigvee\)-semilattice. If X is a Wilker normal space and U(X,Y) is the set of point-closed upper semi-continuous multifunctions equipped with the compact-open topology, then (U(X,Y),\(\subset)\) is a topological lattice.