Two examples concerning hyperspace retraction (Q1107160)

For any continuum X let C(X) be the hyperspace of subcontinua of X, with the Hausdorff metric. A function r: C(X)\(\to X\) is a selection on C(X) if and only if r is continuous and r(A)\(\in A\) for every \(A\in C(X)\). The author is interested in relationships between the following conditions on X: (1) X is a strong deformation retract of C(X), (2) X is a deformation retract of C(X), (3) X is a retract of C(X), (4) C(X) admits a selection. Evidently (1)\(\Rightarrow (2)\Rightarrow (3)\) and (4)\(\Rightarrow (3)\). Two examples constructed in the paper show that \((4)\nrightarrow (2)\) (thus also \((3)\nrightarrow (2)\) and \((4)\nrightarrow (1))\) and \((3)\nrightarrow (4)\).