## $n$-Arc connected spaces

### Volume 130 / 2013

Colloquium Mathematicum 130 (2013), 221-240
MSC: Primary 54F15; Secondary 54D05, 54F15, 54H05.
DOI: 10.4064/cm130-2-5

#### Abstract

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 $\aleph _0$-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.