Span of subcontinua (Q2850655)

The paper under review considers span of a ray \(Y\) limiting to a continuum \(X\). Note that if \(X\) is a simple closed curve then its span is the same as the span of \(Y\). The author proves a somewhat intuitive estimate for cases when \(X\) is not a simple closed curve. Namely, the span of \(Y\) does not exceed the span of \(X\) or the surjective semispan of \(X\), whichever is bigger. Building on the work of \textit{Z. Waraszkiewicz} [``Une famille indenombrable de continus plans dont aucun n'est l'image continue d'un autre'', Fundamenta 18, 118--137 (1932; JFM 58.0632.02)] the author demonstrates the existence of an uncountable family of pairwise incomparable continua for each member of which the set of values of spans of its subcontinua coincides with the Cantor set. A similar result is obtained when the Cantor set is replaced by an arbitrary closed subset of [0,1] containing 0 and 1.