Spaces of lower semicontinuous set-valued maps I

Abstract

Abstract We introduce a lower semicontinuous analog, L −(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L −(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L −(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L −(X) and L −(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L −(X) and L −(Y) can be characterized by a unique factorization.