Unusual space-filling curve 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?
 A: André's answer is correct, but what you really are looking for (I think) is this example, due to Katok, but explained (beautifully) by Milnor.

There is a family of disjoint smooth real-analytic curves $\Gamma_{\beta}$ that fill the unit square $I^2$, and a subset $E \subset I^2$ of full measure, so that each curve $\Gamma_{\beta}$ intersects $E$ in at most one point.

This means that $E$ can be constructed by choosing one point for each parameter $\beta$, so in order not to contradict Fubini, the dependence of $\Gamma_{\beta}$ on $\beta$ cannot be continuous.
A: 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!
