Openly homogeneous continua having only arcs for proper subcontinua
The author proves that if a non-degenerate metric continuum is openly homogeneous and all its non-degenerate proper subcontinua are arcs, then it is a solenoid. Recall that a space is called homogeneous if for each pair of its points there exists a homeomorphism of the space onto itself carrying one point of the pair to the other; it is called openly homogeneous if this condition is weakened to the requirement that there exists an open (continuous) mapping of the space onto itself having above property. The result of the author strengthens a result of \textit{C. L. Hagopian} [Pac. J. Math. 68, 425-435 (1977; Zbl 0364.54027)] who got the same conclusion assuming homogeneity instead of open homogeneity. The so called \(\epsilon\)-push property of homogeneous continua, a corollary of the Effros theorem, played a crucial role in obtaining this result. The author uses an anlogue of the \(\epsilon\)-push property due to \textit{J. J. Charatonik} and \textit{T. Maćkowiak} [Trans. Am. Math. Soc. 298, 579-602 (1986; Zbl 0608.54012)]. An original tool due to the author is introducing a metric \(\hat d\) associated in a special manner with the given metric d. If the continuum is openly homogeneous and all its non- degenerate proper subcontinua are arcs this metric is equivalent to the given one. However, the difference between d and \(\hat d\) is so essential that if they are equivalent and the continuum has only arcs as its non- degenerate proper subcontinua, then the continuum is a solenoid. This is the main lemma of the paper. The author considers also continua whose non-degenerate proper subcontinua are arcs if they are sufficiently small. For continua from this wider class the author proves that they are solenoids, but now the condition under which this is true is somewhat stronger, namely the homogeneity with respect to open light mappings. The problem of omitting ``light'' in the last statement remains open, although if the assumption of lightness is changed by the assumption of the equivalence of metrics d and \(\hat d,\) the statement remains true.
scientific article; zbMATH DE number 4094361
scientific article