Topologically, there is not much choice for orientable closed surfaces: it is a sphere, a doughnut, or a long doughnut with many holes. But if you stay within the 3-dimensional space, the doughnut may form a knot (or multiple holes may pass through each other) with no way of untangling. Not to mention pathological things like this.
In any case, I insist it is more fruitful to state things as fluxes/circulations being unchanged under boundary-preserving defrormation unless charges are crossed. That way the question of "what is a closed orientable surface" does not arise. tens of millions of people stand to see their lives ruined because the bureaucrats at the ECB don't understand introductory economics -- Dean Baker
