The space of selections of smooth fans (Q616916)

The authors show that the space of selections of a smooth fan is a complete separable absolute retract and that it is homeomorphic to the Hilbert space \(\ell_2\). Explicitly, the authors give the following results. A dendroid is an arcwise connected and hereditarily unicoherent continuum. A fan is a dendroid with only one ramification point. A fan \(X\) is smooth provided that there is a point \(p\in X\), such that for each convergent sequence \(\{x_n\}\) in \(X\), converging to a point \(x\in X\), the sequence of arcs \(\{px_n\}\) converges to the arc \(px\) with respect to the Hausdorff metric on \(C(X)\). A selection is a map \(s:C(X)\to X\), such that \(s(A)\in A\) for every \(A\in C(X)\). Given a dendroid \(X\), \(\Sigma{(X)}\) denotes the space of selections of \(X\). The Hilbert space \(\ell_2\) is the vector space defined by \(\ell_2=\{(a_n)_{n=1}^{\infty} \;| \;a_n\in \mathbb R, \sum_{n=1}^{\infty}a_n^2<\infty\}\). \(AR\) is defined to be the class of metric continua which are absolute retracts. Theorem. Let \(X\) be a smooth fan. Then \(\Sigma{(X)}\) is topologically convex. Theorem. Let \(X\) be a smooth fan. Then \(\Sigma{(X)}\) is an AR, separable complete topological space. Theorem. Let \(X\) be a smooth fan. Then \(\Sigma{(X)}\) is homeomorphic to the Hilbert space \(\ell_2\).