\(2^{{\aleph}_0}\) ways of approaching a continuum with \([1,\infty)\) (Q260523)

This is a paper about the number of ``topologically different'' ways you can realise a metric continuum \(X\) as the remainder of a compactification of the ray \([0,\infty)\). It is an old problem, dating back to \textit{Z. Waraszkiewicz} [Fundam. Math. 18, 118--137 (1932; Zbl 0004.22602)], who produced a family of \(\mathfrak{c}:= 2^{\aleph_0}\) pairwise incomparable compactifications of the ray, with \(X\) a simple closed curve. (The ``Waraszkiewicz spirals.'' Here, saying two spaces are \textit{incomparable} means that neither is a continuous image of the other.) Over the years other examples of remainder continua \(X\) were added (e.g., arcs, simple triods); now it is known that any nondegenerate metric continuum may serve, as long as it is locally connected. Without local connectedness, however, the result may no longer be true: any two compactifications of the ray with a pseudo-arc as remainder are comparable [\textit{A. Illanes} et al., Houston J. Math. 41, No. 4, 1325--1340 (2015; Zbl 1348.54027)]. But when the definition of ``topologically different'' is weakened from \textit{incomparable} to \textit{distinct} (i.e., not homeomorphic), large families of compactifications are again forthcoming. In [\textit{V. Martínez-de-la-Vega} and \textit{P. Minc}, Topology Appl. 173, 28--31 (2014; Zbl 1297.54051)], it is shown that for any nondegenerate metric continuum \(X\), there is a family \(\mathcal{K}\) of uncountably many pairwise distinct compactifications of the ray, with \(X\) as remainder. In the present paper it is shown that \(\mathcal{K}\) may be taken to be of cardinality \(\mathfrak{c}\). Indeed, given \(X\), one explicitly constructs a metric compactum \(K\) whose component space, a Cantor set, is such a family \(\mathcal{K}\). It is left as an open question whether \(K\) can be constructed so that every compactification of the ray with remainder \(X\) is homeomorphic to a component of \(K\).