Let X = (X, d] be a metric space. We endow the hyperspace S^X x R consisting of non-empty closed subsets of X x R with the topology induced by d_H defined by d_H(E,F) = inf{epsilon is an element of (0,infinity] | N(E,epsilon) is a subset of F and N(F,epsilon) is a subset E}. Let USCC(X) be a space of upper semi-continuous multi-valued functions phi : X --> R such that phi (x) is a closed interval for every x is an element of X. Identifying those functions with their graphs, we consider USCC(X) as a subspace of 2^X x R. We give a necessary and sufficient condition on X is order that USCC(X) is closed in 2^X x R. In case X is complete, we also give a necessary and sufficient condition on USCC_B(X) to be an AR, where USCC_B(X) is a subspace of USCC(X) consisting of all bounded functions. As a corollary, we find that USCC(X) is an AR if X is compact.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.