Meshed continua have unique second and third symmetric products

Let X be a metric continuum and a positive integer n, we consider the n  -th symmetric product Fn(X)Fn(X) with the Hausdorff metric. In this paper we prove that if X is a meshed continuum, Y   is a continuum and Fn(X)Fn(X) is homeomorphic to Fn(Y)Fn(Y), then X is homeomorphic to Y  , for each n∈{2,3}n∈{2,3}. This answers a question by Alejandro Illanes.