Graph-like continua, augmenting arcs, and Menger’s theorem

Abstract

We show that an adaptation of the augmenting path method for graphs proves Menger's Theorem for wide classes of topological spaces. For example, it holds for locally compact, locally connected, metric spaces, as already known. 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 not containing it by a finite set, in particular to every subspace of the Freudenthal compactification of a locally finite, connected graph. While closed subsets of such a space behave nicely in that they are compact and locally connected (and therefore locally arcwise connected), the general subspaces do not: They may be connected without being arcwise connected. Nevertheless, they satisfy Menger's Theorem.

Cycle spaces in topological spaces

“…The proof consists of showing P is an ab-path in the following sense: if a and b are the ends of e 0 , then P is a connected set containing a and b and, for every other point x of P , a and b are in different components of P \ x. By the comment following Corollary 2.3 in [18], P is locally connected. (A compact totally disconnected space, such as X − E, is 0-dimensional [4,Thm.…”

“…(A compact totally disconnected space, such as X − E, is 0-dimensional [4,Thm. 4.A.11], so (X, E) is a graph-like space as defined in [18].) By Theorem 2.3.17 of [19], P is compact (the ab-prepaths in [19] are the ab-paths as defined here).…”

Eulerian edge sets in locally finite graphs

“…We note that it can also be obtained from a more general result by Thomassen and Vella [11], who prove a Menger-type theorem for graph-like spaces.…”

Embedding a graph-like continuum in a surface

“…For completeness, we give our version of the details here. We remind the reader that a surface and a graph-like continuum are both locally connected (see [15,Thm. 2.1] for the graph-like continuum version); these two facts are used implicitly throughout this section.…”

“…(A space V is 0-dimensional if, for each u, w ∈ V , there is a separation (U, W ) of V with u ∈ U and w ∈ W ; in particular, a 0-dimensional space is totally disconnected and the converse holds for compact spaces.) These spaces are introduced by Thomassen and Vella [15] who prove that a compact, connected, metric graph-like space (a graph-like continuum) is locally connected. They also prove a form of Menger's Theorem for these spaces.…”