Certain finite dimensional maps and their application to hyperspaces

In [8] Y. Sternfeld and this author gave a positive answer to the following longstanding open problem: Is the hyperspace (= the space of all subcontinua endowed with the Hausdorff metric) of a 2-dimensional continuum infinite dimensional? This result was improved in [9] where it was shown that for every positive integer number n a 2-dimensional continuum contains a 1-dimensional subcontiuum with hyperspace of dimension ≥ n. And it was asked there: Does a 2-dimensional continuum contain a 1-dimensional subcontinuum with infinite dimensional hyperspace? In this note we answer this question in the positive. Our proof applies maps with the following properties. A real valued map f on a compactum X is called a Bing map if every continuum that is contained in a fiber of f is hereditarily indecomposable. f is called an n-dirnensional Lelek map if the union of all non-trivial continua which are contained in the fibers of f is n-dimensional. It is shown that for dim X = n + 1 the maps which are both Bing and n-dimensional Lelek maps form a dense G_δ-subset of the function space C(X, double-struck sign I ).