Mapping invariance of extremal continua (Q1186157)

Let \(X\) be a continuum i.e., a compact connected metric space. If \(S\) is terminal in \(I\cup S\) for every irreducible subcontinuum \(I\) in \(X\) such that \(I\cap S\neq\emptyset\neq I\backslash S\), then \(S\) is called an extremal continuum in \(X\). \textit{M. A. Owens} [Topology Appl. 23, 263-270 (1986; Zbl 0598.54016)] defined and gave some characterizations of extremal continua. The aim of this paper is to determine various conditions, concerning the domain \(X\) and/or the continuous mapping \(f\) on \(X\), under which the concept of an extremal continuum is preserved. Among the many results, it is established that if \(X\) is arclike, \(f\) is a monotone mapping defined on \(X\), and \(S\) is an extremal continuum in \(X\), then \(f(S)\) is an extremal continuum in the range.