A characterization of the arc by means of the \(C\)-index of its semigroup (Q688991)

A local dendrite is a Peano continuum \(X\) such that any point \(x \in X\) has a neighbourhood which is a dendrite (i.e. a Peano continuum with no simple curves) and a point \(p\) is a branch point of \(X\) if \(p\) has a dendrite neighbourhood \(D\) such that \(D\setminus \{p\}\) consists of more than two components. A semigroup \(S\) has the \(C\)-index \(C(S) = n\) if \(n\) is the least integer such that every countable subset of \(S\) is contained in a subsemigroup of \(S\) with \(n\) generators. The author characterizes the arc within the class of local dendrites with finitely many branch points as follows: Let \(X\) be a local dendrite with finitely many branch points, \(S(X)\) be the semigroup of all continuous selfmaps of \(X\). Then \(X\) is an arc if and only if the \(C\)-index of \(S(X)\) is two.