n-Arc Connected Spaces

A space is `n-arc connected’ (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed sub-interval of the real line, and (iv) one of a finite family of graphs. General continua that are aleph_0-ac are characterized. The complexity of characterizing n-ac graphs for n=2,3,4,5 is determined to be strictly higher than that of the stated characterization of 7-ac graphs.

💡 Research Summary

The paper introduces a new topological property called n‑arc‑connectedness (n‑ac): a space X is n‑ac if every subset of X consisting of at most n points can be embedded in a single arc (i.e., the image of a closed interval under a continuous injection). This notion generalises the classical concept of arc‑connectedness (the case n = 2) and provides a hierarchy of increasingly restrictive conditions as n grows.

Main Results for Graphs
The authors focus first on one‑dimensional continua that are graphs (finite 1‑dimensional simplicial complexes). They prove that for graphs the following four statements are equivalent:

1. 7‑ac – every set of at most seven points lies on an arc.
2. n‑ac for all n – the space is n‑ac for every natural number n.
3. Injective continuous image of a closed interval – the graph is homeomorphic to the image of a continuous injective map f :