# A special connected subset of the Cantor fan

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, 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 to construct an example. For if $$X$$ is an example then for every two dense connected subsets $$X_1,X_2\subseteq X$$ we must have $$|X_1\cap X_2|>1$$. Essentially the only example I know like this comes from Miller's biconnected set, which needs the Continuum Hypothesis.