Some functions on hyperspaces of continua
For a metric continuum X let \(2^ X\) denote the space of nonempty closed subsets of X with the Hausdorff metric and C(X) the subspace of \(2^ X\) consisting of all nonempty subcontinua of X. In 1977 Goodykoontz examined two functions f,g: \(2^ X\to C(X)\) defined by \[ f(A)=\cap \{K\in C(X):\quad A\subseteq K\}\quad and\quad g(A)=\cap \{K\in C(X):\quad A\subseteq int K\}. \] In this paper the author defines two similar functions F,G: \(2^ X\to C(C(X))\) by \[ F(A)=\{K\in C(X):\quad A\subseteq K\}\quad and\quad G(A)=\{K\in C(X):\quad A\subseteq int K\}. \] Preliminary results include various continuity properties of these functions and their connections with Goodykoontz's functions are examined. These results are then used to characterize smoothness, local connectedness, connectedness im kleinen and the property of Kelley of a continuum.
scientific article; zbMATH DE number 3947159