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

Abstract:

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