Spaces of lower semicontinuous set-valued maps I

Year, pages: 2010, 521 - 540

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.

Mccoy, R. 2010. Spaces of lower semicontinuous set-valued maps I. In Mathematica Slovaca, vol. 60, no.4, pp. 521-540. 0139-9918. DOI: https://doi.org/10.2478/s12175-010-0030-x

Mccoy, R. (2010). Spaces of lower semicontinuous set-valued maps I. Mathematica Slovaca, 60(4), 521-540. 0139-9918. DOI: https://doi.org/10.2478/s12175-010-0030-x