  Let $X$ be a Peano continuum having a free arc and let $C^0(X)$ be the
semigroup of continuous self-maps of $X$. A subsemigroup $F\subset C^0(X)$ is
said to be sensitive, if there is some constant $c>0$ such that for any
nonempty open set $U\subset X$, there is some $f\in F$ such that the diameter
${\rm diam}(f(U))>c$. We show that if $X$ admits a sensitive commutative
subsemigroup $F$ of $C^0(X)$ consisting of continuous open maps, then either
$X$ is an arc, or $X$ is a circle.
