Is there a dense connected subset $X$ of the Cantor fan $$(C\times [0,1])/(C\times \{1\})$$ such that for every two connected subsets $X_1,X_2\subseteq X$, the intersection $X_1\cap X_2$ is connected?

This would produce a counterexample to Problem 76 in the [Houston Problem Book][1], which is still open I think.

Note that $X$ minus the vertex point must be totally disconnected, having only one point in each blade of the fan. I constructed a connected set with this property by taking the complete Erdös space in the Cantor fan, and moving its points up and down the blades using a sin function to make it dense.

The Continuum Hypothesis may be necessary.


  [1]: http://hyperspacewiki.org/index.php/The_Houston_Problem_Book_Problems_51-100