On the arc-wise connection relation in the plane (Q6559427)

\textit{K. Kunen} and \textit{M. Starbird} [Topology Appl. 14, 167--170 (1982; Zbl 0504.54019)] constructed a compact connected set \(K\) in \(\mathbb R^3\) having a non Borel arc-wise component and they asked whether such set can be constructed in \(\mathbb R^2\). The main result of the paper under review gives a negative answer to this question.\N\NIn fact the authors prove that all arc-wise connected components \(C\) of any \(G_\delta\) set \(X\) in the plane are Borel. Together with Proposition 5.1 of \textit{H. Becker} and \textit{R. Pol} [Topology Appl. 114, No. 1, 107--114 (2001; Zbl 0997.54057)] it even gives that the arc-wise connection relation of \(X\) is Borel, which is the main result of the paper.\N\NConcerning the method of proof let us point out that an appropriate topology, on each \(G_\delta\) set \(X\) in the plane, which refines the standard topology on \(X\) serves as a tool in the proof of the fact that each arc-wise connected component \(C\) of \(X\) containing a simple Moore triod is Borel.