Suppose that $C\subseteq [0,1] $ is a Cantor set (i.e. a totally disconnected closed perfect set) and $F\subseteq C \times [0,1]$ is a closed set such that the following holds: For every clopen-in-$C$ set $Q$, the set $F(Q)=\{y|(\exists x\in Q )((x,y)\in F)\}\subseteq [0,1]$ satisfies $Q\subseteq F(Q)^\circ$, where $A^\circ$ is the interior of $A$.
Does it follow that there is some $x\in C$ such that $F(x)=\{y|(x,y)\in F\}$ has non-empty interior? If it helps at all in the specific case I care about $F$ comes from some closed $G\subseteq [0,1]^2$ which is symmetric around the diagonal.
The idea of the title is that this is similar to the version of the Baire category theorem that says that if a countable family of closed sets covers a complete metric space, then one of the closed sets has non-empty interior ($F(C)$ doesn't cover $[0,1]$, but it does cover certain sub-intervals).
The implication certainly fails if you drop the requirement with the interiors. We can take $C$ to be the standard one-thirds Cantor set and $F$ to be the graph of a continuous surjection of $C$ onto $[0,1]$, for instance.