Semi-terminal continua in homogeneous spaces (Q2828642)

This lengthy paper is devoted to the study of semi-terminal continua in homogeneous metric spaces. The concept of semi-terminality was defined by the author in [Can. J. Math. 62, No. 1, 182--201 (2010; Zbl 1187.54028)]. The current work builds on the author's paper on semi-terminal continua in Kelley spaces [Trans. Am. Math. Soc. 363, No. 6, 2803--2820 (2011; Zbl 1226.54035)] and his joint paper with \textit{K. Whittington} [Topology Appl. 154, No. 8, 1581--1591 (2007; Zbl 1119.54025)]. The concept of a Kelley space dates back to 1940. A space is called a Kelley space if for each sequence \(\{x_n\}\) of points in \(X\) convergent to a point \(x\) in a subcontinuum \(K\) of \(X\) there exists a sequence \(\{K_n\}\) of continua convergent to \(K\) such that each \(x_n\) is contained in the respective continuum \(K_n\) [\textit{J. L. Kelley}, Trans. Am. Math. Soc. 52, 22--36 (1942; Zbl 0061.40107)]. The author gives an extensive review of prior results established for Kelley spaces as related to terminal continua, followed by a review of the prior work on semi-terminal and semi-decomposable continua. The main thrust of the paper, however, is the position of a semi-terminal continuum in a homogeneous continuum. Recall that a subcontinuum \(T\) of a space \(X\) is called terminal in \(X\) if for every subcontinuum \(Y\) of \(X\) that has a nonempty intersection with \(T\) either \(T\) is contained in \(Y\) or vice versa. A subcontinuum \(T\) of a space \(X\) is called semi-terminal if for each pair of disjoint subcontinua \(Z_1\) and \(Z_2\) of \(X\), each of which has a nonempty intersection with \(T\), either \(Z_1\) or \(Z_2\) is a subset of \(T\). The author calls a continuum \(X\) semi-indecomposable when each two subcontinua of \(X\) with nonempty interiors have a nonempty intersection, pointing out that it is the same concept as that of a strictly non-mutually aposyndetic continuum [\textit{C. L. Hagopian}, Proc. Am. Math. Soc. 23, 615--622 (1969; Zbl 0184.26603)]. It is proven that if \(X\) is a homogeneous continuum and \(Y\) is a proper semi-terminal subcontinuum of \(X\) then \(Y\) is terminal if and only if it is indecomposable. Furthermore, every proper semi-terminal subcontinuum of a homogeneous continuum has empty interior. The main result -- the semi-terminal decomposition theorem -- demonstrates that a homogeneous continuum has the finest monotone, upper semi-continuous decomposition into semi-terminal homogeneous subcontinua so that the quotient is a homogeneous continuum without proper, nondegenerate semi-terminal subcontinua. In [Zbl 1119.54025], the author together with \textit{K. Whittington} defined, for a space \(X\), a filament in \(X\) as a subcontinuum \(K\) of \(X\) possessing a neighbourhood \(N\) whose component containing \(K\) has empty interior. In the paper under review, the semi-terminal and filament index \textbf{stf}, with values \(-1\), \(0\) or \(1\), is assigned to every homogeneous continuum \(X\) based on the semi-terminal components of \(X\) being properly contained in the filament components of \(X\), being identical with the filament components of \(X\) or properly containing the filament components of \(X\), respectively. Homogeneous continua with \textbf{stf} equal to \(0, -1\) or \(1\) each have a section devoted to them. The paper concludes with several open questions related to the work of Rogers on cell-like decomposition [\textit{J. T. Rogers jun.}, Proc. Am. Math. Soc. 87, 375--377 (1983; Zbl 0504.54037)].