topological transversal and differential transversal Given a smooth vector fied on the two-torus, suppose it has a topological transversal, i.e. there exists a closed curve that intersects each integral curve of the vector field, does it imply that there exists a differential transversal, i.e. there exists a $C^1$ closed curve that is transverse to every integral curve? 
 A: Since you are on the 2-torus, I guess that you assume that your vector field does not vanish. You seem to give a definition of "topological transversal"; your definition is too large; there is always a closed continuous Peano curve filling the 2-torus! And there is always a smooth simple closed curve $C$ meeting every integral curve.
There is a known notion of "topological transversality"; it means that at every point $x$ of your curve $C$ there is a local topological chart
$\phi:U(\subset T^2)\to V(\subset R^2)$ sending $x$ to $0$ and $C$ to $R\times 0$ and the integral curve through $x$ to $0\times R$. With this definition, the existence of a topological total closed transversal implies the existence of a $C^1$ simple closed curve transverse (in the usual sense) to the vector field and meeting every integral curve. Indeed, both are equivalent to the absence of "plane Reeb components".
A: If you assume that the closed curve does not intersect any zero of the vector field, the answer is yes, I think. Just cover your closed transversal by a finite number of flow boxes, and build a smooth transversal, successively in each flow box.
