
doi: 10.1007/bf02844332
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.
hereditarily unicoherent, Continua and generalizations, Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces, weak cutpoint order, continuum, Unicoherence, multicoherence, irreducible
hereditarily unicoherent, Continua and generalizations, Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces, weak cutpoint order, continuum, Unicoherence, multicoherence, irreducible
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