An uncountable family of copies of a non-chainable tree-like continuum in the plane (Q2839325)

A triod is a continuum \(X\) having a point \(v\) and three subcontinua \(A_1\), \(A_2\) and \(A_3\) such that each \(A_i\) is irreducible between \(v\) and other point and the only point in the intersection of \(A_i\) and \(A_j\) is \(v\) if \(i\) is different from \(j\). A classical and useful result proved by \textit{R. L. Moore} [Proceedings USA Academy 14, 85--88 (1928; JFM 54.0630.03)] says that the plane does not contain an uncountable family of pairwise disjoint triods. It is known that if \(X\) is a triod and \(S\) is any infinite compact space, then \(X\times S\) cannot be embedded in the plane. In this direction, Fugate asked whether \(X\times S\) can be embedded in the plane if \(X\) is a continuum which is neither chainable nor circle-like and \(S\) is the space that consists of a sequence and its limit. \textit{W. T. Ingram} [Fundam. Math. 85, 73--78 (1974; Zbl 0281.54014)] produced an example of an uncountable family of pairwise disjoint non-chainable tree-like continua in the plane, thus demostrating that Moore's theorem cannot be generalized to non-chainable tree-like continua. However, the continua in Ingram's collection are not mutually homeomorphic.NEWLINENEWLINEIn the paper under review, the author constructs a non-chainable tree-like continuum \(X\) such that the product of \(X\) with the Cantor set can be embedded in the plane. The continuum \(X\) has the property that every proper subcontinuum of \(X\) is an arc. The paper includes the following interesting question.NEWLINENEWLINEQuestion. Is there a hereditarily indecomposable non-chainable tree-like continuum \(X\) such that the plane contains an uncountable collection of pairwise disjoint copies of \(X\)?