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.

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/MAfKd.png
  [2]: https://i.sstatic.net/4OfJR.png

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.