Graph-like continua, augmenting arcs, and Menger's theorem (Q2390152)

Menger's theorem has been a cornerstone of graph theory, since 1930. The authors show that a variant of the augmenting path method used for finite graphs proves Menger's theorem for wide classes of topological graphs. For example, it is easily proved for locally compact, locally connected, metric spaces, as already known by results of \textit{G. I. Whyburn} [Trans. Am. Math. Soc. 63, 452--456 (1948; Zbl 0031.41802)]; moreover, in virtue of \textit{G. Nöbeling} [Fundam. Math. 18, 23--38 (1932; Zbl 0004.16204)], the method applies to the so called graph-like continua. The method lends itself particularly well to another class of spaces, namely the locally arcwise connected, hereditarily locally connected, metric spaces. Finally, it applies to every space where every point can be separated from every closed set non containing it by a finite set, in particular to every subspace of the Freudenthal compactification of a locally finite, connected graph.