Vol. 19, No. 3, 1966

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Closed and image-closed relations

Stanley Phillip Franklin and R. H. Sorgenfrey

Vol. 19 (1966), No. 3, 433–439

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 Tand T, where Tis 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 Tor 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.

Mathematical Subject Classification
Primary: 54.23
Received: 13 August 1965
Published: 1 December 1966
Stanley Phillip Franklin
R. H. Sorgenfrey