If the square $[0,1]^2$ is covered by two closed sets $A\subseteq [0,1]^2$, $B\subseteq [0,1]^2$ with $[0,1]^2 = A \cup B$, is there always a path inside $A$ from lower edge $[0,1]\times \{0\}$ to the upper edge $[0,1]\times \{1\}$ or a path inside $B$ from the left edge $\{0\}\times[0,1]$ to the right edge $\{1\}\times[0,1]$?
I struggled to find a proof or counterexample by myself and online. I suspect it is true, because it does not seem possible to 'block' all paths from the bottom to the top with the set $B$ without simultaneously creating a path from left to right.
If the sets $A,B$ needn't be closed, as considered in this question, it is possible that no path exists, for example by using the topologists sine curve as done here.
If both sets were open instead of closed, I believe one can show the existence of a path by proceeding roughly similar to this: Choose a grid size $\delta$ significantly smaller than a Lebesgue number $\epsilon$ and overlay a square grid of that mesh size it on the square. Color a grid point red if a $\frac{\epsilon}{2}$-ball around the point lies completely in A, otherwise the ball must be completely in B and we color it blue. Then by the Steinhaus chessboard theorem (or by the hex theorem if we add a diagonal connection to each mesh square) there is a path connecting opposite sides in one color and it must be completely in $A$ or $B$.
In this paper, Gale also proves an $n$-dimensional version of the hex theorem, so additionally it would be interesting to see what happens in $n$-dimensions: If $[0,1]^n$ is covered by $n$ closed sets $C_1,\dots,C_n$, is there an $i$ such that there is a path in $C_i$ from the $\{x\in[0,1]^n|x_i=0\}$ face to its opposite side $\{x\in[0,1]^n|x_i=1\}$?
Because the KKM lemma can be proven by applying the discrete version, Sperners lemma, for smaller and smaller grids, I tried repeatedly using the hex theorem, but this didn't lead anywhere for me
