It is not known if there is a 2-to-1 map from a continuum onto a tree-like continuum. In fact, it is not known if there is a 2-to-1 map onto a hereditarily decomposable tree-like continuum. We show that the domain of such a map would have to contain an indecomposable continuum.
Copyright © 2004 American Mathematical Society. This article first appeared in Proceedings of the American Mathematical Society 132:1 (2004), 283-289.
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Heath, Jo and Van C. Nall. "Tree-like Continua and 2-to-1 Maps."Proceedings of the American Mathematical Society 132, no. 1 (2004): 283-89.