Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure.

EDIT: The answer below is unsatisfactory, and possibly incorrect. It does not account for the path-component of the main orbit being enlarged when the the closure is taken. In the forking regions there will be a Cantor set times "T", with one leg of "T" butting into the main orbit.

exercise. You should prove it. (If you are having trouble, post it on MSE.) Note the assumption that this is adenseopen set. A flowline of a differential equation (which exists for all time) is always diffeomorphic to either $S^1$ or $\Bbb R$, depending on whether or not the flowline is periodic. It is not, in this case. Therefore the flowline that you take the closure of is diffeomorphic to $\Bbb R$. $\endgroup$ – Mike Miller Jan 22 at 22:59