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A well known theorem of R. L. Moore states that the plane does not contain an uncountable family of pairwise disjoint triods. In 1974, Ingram demonstrated that the same is not true for non-chainable tree-like continua. The continua in Ingram’s family are not pairwise homeomorphic, making the example less applicable to the study of homogeneous continua in the plane. In this paper, we construct a non-chainable tree-like continuum X such that the product of X with the Cantor set can be embedded in the plane.

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