Spaces of upper semi-continuous multi-valued functions on separable metric spaces

Abstract

Let $X=(X, d)$ be a metric space. By USCC(X, I), we denote the space of upper semi-continuous multi-valued functions $\varphi:X\rightarrow I=[0,1]$ such that each $\varphi(x)$ is a closed interval. Each $\varphi\in USCC(X, I)$ can be identified with its graph, which is a closed subset of $X\times I$. The space USCC(X, I) admits the Hausdorff metric induced by the product metric on $X\times I$. In this paper, by proving the converse of Fedorchuk’s result, we show that USCC(X, I) is homeomorphic to the Hilbert cube $Q=[-1,1]^{\omega}$ if and only if $X$ is infinite, locally connected and compact. In case $X$ is a dense subset of a locally connected metric space $Y$ such that $Y\backslash X$ is locally nonseparating in $Y$, USCC $(X, I)$ can be regarded as a subspace of USCC $(Y, I)$. It is also proved that the pair (USCC $(Y, I)$ , USCC $(X, I))$ is homeomorphic to $(Q, s)$ if and only if $X\neq Y,$ $X$ is $G_{\delta}$ in $Y$, and $Y$ is compact, where $s=(-1,1)^{\omega}\subset Q$.