Iterated quasicomponents of subspaces of rational continua (Q689594)

A space \(X\) is called rational if it has a base of open sets each having a boundary which is countable. For any \(x\in X\) define the index of nonconnectedness of \(X\) at \(x\), \(nc(X,x)\), to be the smallest ordinal \(\alpha\) such that the component of \(x\) at \(X\) is obtained by iterating the taking of the quasicomponents at \(x\), \(\alpha\) many times. \textit{A. Lelek} [Fundam. Math. 67, 359-367 (1970; Zbl 0201.248)] has proved that if \(X\) is a rational metrizable continuum, then \(\sup\{nc(Y,x): x\in Y\}<\omega_ 1\) for all \(Y\subset X\) and has asked if this can be improved to \(\sup\{nc(Y,x): x\in Y\subset X\}<\omega_ 1\). In this article, the author has constructed a rational planar continuum \(X\) which is the countable union of straight-line segments having a common end- point and with a point \(x\) for which \(\sup\{nc(Y,x): x\in Y\subset X\}= \omega_ 1\).