Problem 1 (HPB): Suppose $X$ is a continuum which cannot be embedded in any continuum $Y$ such that $Y$ is the union of a countable family of arcs. Does $X$ contain a connected subset that is not path connected? (Asked by E.D. Tymchatyn, 09/30/81)

Answer: No (E.D. Tymchatyn, 09/30/81).