Ultra smooth continua are smooth (Q1809695)

A continuum \(X\) is said to be smooth at a point \(p \in X\) if it is hereditarily unicoherent at \(p\) and for each \(x \in X\) and each sequence \(\{x_n\}\) converging to \(x\) the sequence of irreducible continua \(\{I(p,x_n)\}\) converges to the irreducible continuum \(I(p,x).\) \(X\) is said to be smooth if it is smooth at some point. A continuum \(X\) is said to be ultra smooth at a point \(p \in X\) if it is hereditarily unicoherent at \(P\) and for every pair of points \(x\) and \(y\) of \(X\) there exists a retraction of \(X\) onto the subcontinuum \(Y = I(p,x) \cup I(P,y)\) preserving the weak cutpoint order. The author shows that if a continuum \(X\) is ultra smooth at a point \(p\) then it is smooth at \(p.\) It is known by an example of \textit{Lewis Lum} [Stud. Topol., Proc. Conf. Charlotte, N.C., 1974, 331-338 (1975; Zbl 0306.54050)] that the converse is false even for dendroids. The author also characterizes ultra smooth continua which are formed as the union of a null sequence of irreducible continua.