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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 subinterval of the real line, and (iv) one of a finite family of graphs. General continua that are ℵ₀-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.
A hit-and-miss topology () is defined for the hyperspaces , Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between and the Vietoris topology and we find conditions on X for which these topologies are equivalent.
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