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Abstract

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 $T_n$ with $dim C (T_n) ≥ n$. This implies that X contains a compact one-dimensional subset T with dim C (T) = ∞.