Around 1998, I encountered a (forgotten) reference to a particularly strange space-filling curve.


Consider a foliation as a collection of continuous nonintersecting curves that start at $(0,0)$ and end at $(1,1)$ and collectively fill the unit square, such as the graphs of functions $f\_t(x) = x^t$ where $t \ge 0$. Supposedly there exists a continuous curve G that starts at $(1,0)$, ends at $(0,1)$, fills the unit square, and crosses each $f\_t$ curve only once.


This initially sounds even more impossible than the Cantor curve. But intuitively a space-filling curve could trace back and forth over the $f\_t$ curves and only cross at the corners $(0,0)$ and $(1,1)$. Can someone please explain a construction of such a space-filling curve?



Unusual space-filling curve
The space filling curve you are looking for does not exist.


Assume by contradiction that such a space filling curve $\gamma:I\rightarrow [0,1]^2$ exists.
Since $\gamma$ intersects each curve $f\_t\subset [0,1]^2$ only once, the preimage $\gamma^{-1}(f\_t)$ is either a point or an interval. The curve $\gamma$ being space-filling, that preimage can't be a point. It is therefore an interval and, in particular, of positive measure.


Letting $t$ vary,
we have constructed an uncountable family of disjoint subsets of $[0,1]$, all of whom have positive measure: contradiction!


