Hyperspaces through regular and meager subcontinua
A continuum is a nonempty compact connected metric space. By a hyperspace of a continuum \(X\) we mean a specified collection of subsets of \(X\) endowed with the Hausdorff metric. Given a continuum \(X\), the hyperspace of regular subcontinua \(D(X)\) is the collection of all regular subcontinua of \(X\) and the hyperspace of meager subcontinua \(M(X)\) is the collection of all subcontinua of \(X\) with empty interior. The article under review discusses the following problem: Problem. Characterize the families of continua for which any of the following equalities hold (a) \(F_1(X) = M(X)\); (b) \(D(X) = C(X) \setminus F_1(X)\); (c) \(D(X) = C(X) \setminus M(X)\); (d) \(C(X) = D(X) \cup M(X)\). \medskip Some results of the paper are the following: Theorem. Let \(X\) be a continuum. It is true that (a) If \(M(X) = F_1(X)\), then \(X\) is hereditarily locally connected. \medskip (b) If \(D(X) = C(X) \setminus F_1(X)\), then \(X\) is hereditarily locally connected. \medskip Theorem. Let \(X\) be a locally connected continuum. If \(D(X) = C(X) \setminus M(X)\), then \(M(X) = F_1(X)\). \medskip Theorem. If \(X\) is a hereditarily locally connected continuum, then the following conditions are equivalent: (i) \(X\) does not contain dendrites \(\mathcal{D}_1\); (ii) \(M(X) = F1(X)\); (iii) \(D(X) = C(X) \setminus F_1(X)\); (iv) \(D(X) = C(X) \setminus M(X)\); (v) \(C(X) = D(X) \cup M(X)\); Also, the author presents interesting problems in this line of research. This article contains some interesting results.
scientific article; zbMATH DE number 7387398