Filament sets and decompositions of homogeneous continua
This paper is a continuation of another paper [Topology Appl. 154, 1581--1591 (2007; Zbl 1119.54025)] by the same authors in which they defined filament and ample continua. In the present paper, the authors study decompositions of homogeneous continua using these concepts which generalize Jones's aposyndetic decomposition. A continuum is a compact connected metric space. A continuum \(X\) is homogeneous if for any two points \(x_1\) and \(x_2\) of \(X\), there exists a homeomorphism \(h:X\to X\) such that \(h(x_1)=x_2\). A map \(f:X\to Y\) is atomic provided that for every subcontinuum \(K\) of \(X\) either \(f(K)\) is degenerate or \(K=f^{-1}(f(K))\). A decomposition \(\mathcal{D}\) of a continuum \(X\) is atomic if the quotient map \(\pi:X\to X/\mathcal{D}\) is atomic. A decomposition \(\mathcal{D}\) of a continuum \(X\) is respected by homeomorphisms if \(h(D)\in\mathcal{D}\) for every \(D\in\mathcal{D}\). If \(X\) is a continuum, a subcontinuum \(K\) of \(X\) is called filament (in \(X\)) provided that there exists a neighborhood \(N\) of \(K\) in \(X\) such that the component, \(Cnt_N(K)\), of \(N\) containing \(K\) has empty interior. A subset \(F\) of the continuum \(X\) is a filament set if any continuum \(K\) contained in \(F\) is filament in \(X\). The subcontinuum \(K\) of \(X\) is called ample provided that for every open set \(U\) of \(X\) containing \(K\), there exists a subcontinuum \(L\) of \(X\) such that \(K\subset Int(L)\subset L\subset U\). A continuum \(X\) is filament additive if for each two filament subcontinua \(K\) and \(L\) of \(X\) with nonempty intersection, the union \(K\cup L\) is filament. The paper contains many interesting results. We mention some of them. Proposition 3.2. Let \(f:X\to Y\) be an atomic map. (1) If \(K\) is a subcontinuum of \(X\) such that \(f(K)\) is nondegenerate, then \(K\) is indecomposable if and only if \(f(K)\) is indecomposable. (2) \(X\) is hereditarily indecomposable if and only if \(Y\) is hereditarily indecomposable and \(f^{-1}(y)\) is hereditarily indecomposable for each \(y\in Y\) If \(f:X\to Y\) is an atomic map, then a subcontinuum \(K\) of \(X\) is of Type I if \(f(K)\) is degenerate and \(K\) is of Type II provided that \(K=f^{-1}(f(K))\). Proposition 3.3. Let \(X\) be a homogeneous continuum and \(f:X\to Y\) be an atomic map. Then a subcontinuum \(K\) of \(X\) is filament in \(X\) if and only if either \(K\) is of type I or \(f(K)\) is filament in \(Y\). Theorem 3.6. Let \(X\) be a homogeneous continuum and let \(\mathcal{J}\) be an atomic decomposition of \(X\). Then for each class: (1) filament additive, (2) non-filament additive, (3) filament connected, (4) neither filament additive nor filament connected, (5) indecomposable. \(X\) is a member of this class if and only if \(X/\mathcal{J}\) is, too. Given a homogeneous continuum \(X\), the authors use functions \(f:X\to P(X)\), where \(P(X)\) is the power set of \(X\), to define continuous decompositions of \(X\). They prove some general results and present several examples.
scientific article; zbMATH DE number 5159479
scientific article