suppose curves f and g on a 2d surface embedded in 3d are disjoint except at point p where f(1/2)=g(1/2)=p. we say that f crosses g at p if for all small enough radius r f intersects both components of B(p,r) \ g. where B(p,r) is the ball of radius r about p.
Is this the standard way to defining "cross"
f crosses g iff g crosses f
I imagine one can prove this from the Jordan curve theorem but I wanted to ask if it has a name.
Is a notion of piecewise differentiability required?