A Categorical Characterization of some Bitopological Hyperspaces
by
Bruce S. Burdick
Roger Williams University

Given a bitopological space (X, T, T\sp *) we consider various hyperspaces (H(X), L(T), U(T\sp *)), where L(T) is the lower Vietoris topology generated by T, U( T\sp *) is the upper Vietoris topology generated by T\sp *, and H(X) is some collection of subsets of X. For H(X)=P₀(X), the non-empty subsets of X, there is a straightforward characterization of this hyperspace as the result of applying to (X, T, T\sp *) a functor which is right adjoint to \iota:T --> T\sp m, where T is the category of topological spaces and continuous functions, T\sp m is a category of topological spaces and a certain class of continuous relations, and \iota is inclusion.

There are three other reasonable choices for H(X), each having the advantage of making (H(X), L(T), U(T\sp *)) a T₀ bitopological space. These other choices create obstacles to the categorical characterization above. We discuss modifications to the basic construction above which circumvent these obstacles. Among these modifications is a redefinition of the composition of relations which is due to Frank Oles.

Date received: August 2, 1999