The space of subcontinua of a 2-dimensional continuum is infinite dimensional

Abstract:

Let $X$ be a metric continuum and let $\mathcal {C}(X)$ denote the space of subcontinua of $X$ with the Hausdorff metric. We settle a longstanding problem showing that if $\dim X = 2$ then $\dim \mathcal {C}(X)= \infty$. The special structure and properties of hereditarily indecomposable continua are applied in the proof.