Arc-smooth continuum X admits a Whitney map for C(X) iff it is metrizable

Let X be a non-metric continuum, and C(X) be the hyperspace of subcontinua of X. It is known that there is no Whitney map on the hyperspace 2X for non-metrizable Hausdorff compact spaces X. On the other hand, there exist non-metrizable continua which admit and ones which do not admit a Whitney map for C(X). In particular, locally connected or rim-metrizable continuum admits a Whitney map if and only if it is metrizable. In this paper we will show that an arc-smooth continuum X admits a Whitney map for C(X) if and only if it is metrizable.