TITLE: Unusual Space-Filling Curve
QUESTION [7 upvotes]: 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 ft(x) = xt where t >=0.  Supposedly there exists a continuous curve G that starts at (1,0), ends at (0,1), fills the unit square, and crosses each ft 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 ft curves and only cross at the corners (0,0) and (1,1).  Can someone please explain a construction of such a space-filling curve?

REPLY [5 votes]: 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.