Houston Problem Book Problem 15

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Problem 15 (HPB): Suppose $f$ is an open mapping of a compact metric space $X$ to the Sierpiński curve. Do there exist arbitrarily small closed neighborhoods $U$ of $x$ in $X$ for which $y$ is in $\mathrm{Int}(f(U))$ and $f \big|_U$ is confluent? (Asked by A. Lelek, 24 November 1971)

Answer: No (T. Maćkowiak and E.D. Tymchatyn 25 January 1981)