If X and Y are topological
spaces, a relation T ⊆ X ×Y is upper semi-continuous at the point x of the domain
D(T) of T if for each neighborhood V of T(x), there is a neighborhood U of x such
that T(U) ⊆ V . Results so far published about such relations usually require that
they be closed (as subsets of the product space) or image-closed (T(x) is closed in Y
for each x ∈ X). Given any relation T, it seems natural to consider the associated
relations T′ and T, where T′ is defined by T′(x) =T(x) and T is the closure
of T in the product space. In particular, it is pertinent to ask under what
conditions the upper semi-continuity of T implies that of T′ or T, or that
T′ =T. As might be expected, the answers to these questions take the form of
restrictions on Y , and, indeed, serve to characterize regularity, normality, and
compactness.
