Filament sets and homogeneous continua
The metric continuum \(X\) is called a Kelley continuum provided that whenever a sequence of points \(x_n\) converges to an element \(x\) contained in a subcontinuum \(A\) of \(X\), there are subcontinua \(A_n\) of \(X\) such that \(x_n\) belongs to \(A_n\), for each \(n\), and the sequence \(A_n\) converges to \(A\). \textit{R. W. Wardle} [Houston J. Math. 3, 291--299 (1977; Zbl 0355.54025)] proved that homogeneous metric continua are Kelley continua. A subcontinuum \(A\) of \(X\) is said to be filament provided that it has a neighborhood in which the component containing \(A\) has empty interior. The subcontinuum \(A\) is said to be ample provided that for every open set \(U\) containing \(K\) there is a continuum \(B\) such that \(A\) is contained in the interior of \(B\) and \(B\) is contained in \(U\). The paper under review is the introductory paper in a series of articles which develop and apply new tools in continuum theory based on the concepts of filament and ample subcontinua. These techniques are specially suited for studying homogeneous continua as well as the larger class of Kelley continua. Two samples of the results proved in this paper are the following. Proposition 2.3. Let \(X\) be a Kelley continuum and \(A\) a subcontinuum. The following statements are equivalent: (a) \(A\) is ample, (b) \(A\) is not filament, (c) \(C(X)\) is connected im kleinen. Corollary 3.3. Let \(X\) be a homogeneous continuum. The following conditions are equivalent: (a) \(X\) is not locally connected, (b) Each singleton in \(X\) is filament, (c) Sufficiently small subsets of \(X\) are filament, (d) The topology on \(X\) has a base of open filament sets. The paper ends with a series of interesting questions.
scientific article; zbMATH DE number 5151650
scientific article