### Basis element for the induced topology on a genus two surface

A genus two surface embedded in Euclidean space, shown below [.mov,.avi] $$\Sigma_2 \subset \mathbb{R}^3$$ inherits a topology whose basis elemnts are intersections of the surface with open balls $$\mathcal{B}_i = \Sigma_2 \cap B(\epsilon).$$ A random such intersection is shown below on the left, [.mov,.avi]. The animation on the right [.mov,.avi] shows three different basis elements.