Extending surjections defined on remainders of metric compactifications of \([0,\infty)\) (Q2786844)

\textit{A continuum} is a connected compact space. A continuum is \textit{indecomposable} if it cannot be represented as the union of any two of its proper subcontinua. A continuum is \textit{hereditarily indecomposable} if all of its subcontinua are indecomposable. A continuum \(X\) is \textit{chainable} if for any \(\varepsilon>0\) there exists a sequence \(U_1, \dots,U_n\) of open subsets of \(X\) such that \(\operatorname{diam}(U_i)<\varepsilon\) for each \(i\leq n\) while \(U_1\cup \dots \cup U_n=X\) and \(U_i \cap U_j\neq \emptyset\) if and only if \(|i-j|\leq 1\).NEWLINENEWLINEThe authors prove that for any compactifications \(K_0\) and \(K_1\) of the real half-line \([0,\infty)\), if \(R_0=K_0\setminus [0,\infty)\) is a hereditarily indecomposable continuum and \(R_1=K_1\setminus [0,\infty)\) is a chainable continuum, then every continuous surjection \(f:R_0\to R_1\) can be extended to a continuous surjection \(f^*:K_0 \to K_1\).