A dyadic square is a subset of $R^2$ of the form $x + 2^{n} [0,1]^2$ with $x \in 2^{m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset of $A \cap S$ which intersects two opposite sides of the square $S$. Clearly, the 45 degree line $\{ (\pi + t, t) : t \in R \}$ does not cross any dyadic square. Does every nontrivial closed, connected set that is not a line segment cross some dyadic square?
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