I am asked the following question: (I write $\{a,b\}$ for points in $\mathbb{R}^2$)
Let $\{0,0\},\{1,1\} \notin K \subset [0,1]^2$ such that the projections of $K$ onto the $x$-axis and the $y$-axis are (1 dimensional) lebesgue null-sets. Is there a curve $\gamma : [0,1] \longrightarrow [0,1]^2\backslash K$ such that $\gamma(0)=\{0,0\}, \gamma(1)=\{1,1\}$ and $\ell(\gamma)\leq2$?
My idea was to consider a family of disjunct curves to do a dimension argument, but I stumbled across the cantor-set and therefore saw that $K$ doesn't have to be countable, might there even be a counter example?