Note on path-components in complete spaces (Q5939835)

In any topological space the relation of being in the same path-component is an equivalence relation. The authors prove that this relation is Borel if the underlying space is compact, metrizable and has the property that every simple closed curve intersects only finitely many (other) simple closed curves in more than one point. They also show that for a \(G_\delta\)-set in the plane the relation is Borel iff all path-components are. These results extend and complement results from [\textit{H. Becker}, Lect. Notes Log. 11, 1-16 (1998; Zbl 0894.03029)].