Applications of the $u$-closure operator

Abstract:

Let ${\text {c}}{{\text {l}}_u}(A)$ be the $u$-closure of a subset $A$ of a space. We prove that a space is compact if and only if for each upper-semicontinuous multifunction $\lambda$ on the space, the multifunction $\mu$ defined on the space by $\mu (x) = {\text {c}}{{\text {l}}_u}(\lambda (x))$ assumes a maximal value under set inclusion. We also prove that in a Urysohn-closed space any two subsets with disjoint $u$-closures are separated by disjoint open subsets. The quotient space induced by identifying those points with identical $u$-closures is investigated and shown to be ${T_0}$.