Hyperspaces of two-dimensional continua

Let X be a compact metric space and let C(X) denote the space of subcontinua of X with the Hausdorff metric. It is proved that every two-dimensional continuum X contains, for every n ≥ 1, a one-dimensional subcontinuum ${\displaystyle T_n}$ with ${\displaystyle dimC\left(T_n\right)\ge n}$. This implies that X contains a compact one-dimensional subset T with dim C (T) = ∞.