Cut points of 𝑋 and the hyperspace of subcontinua 𝐶(𝑋)

Abstract

Let X X be a nondegenerate metric continuum and p 0 {p_0} a point with X = X 1 ∪ X 2 X = {X_1} \cup {X_2} , { p 0 } = X 1 ∩ X 2 \{ {p_0}\} = {X_1} \cap {X_2} , X 1 {X_1} and X 2 {X_2} continua. Denote by C ( X ) C(X) , C ( X 1 ) C({X_1}) and C ( X 2 ) C({X_2}) the hyperspaces of nonempty subcontinua of X X , X 1 {X_1} and X 2 {X_2} respectively. Theorem. C ( X ) C(X) is contractible if and only if C ( X 1 ) C({X_1}) and C ( X 2 ) C({X_2}) are contractible and either X 1 {X_1} or X 2 {X_2} is contractible im kleinen at p 0 {p_0} (a modification of connected im kleinen at p 0 {p_0} ). Theorem. Let X 1 {X_1} and X 2 {X_2} satisfy Kelley’s condition K K . Then C ( X ) C(X) is contractible when and only when either X 1 {X_1} or X 2 {X_2} is connected im kleinen at p 0 {p_0} . Examples are given.