Basis element for the induced topology on a genus two surface

A genus two surface embedded in Euclidean space, shown below embedded in \(\mathbb{R}^3\) \[\Sigma_2 \subset \mathbb{R}^3\] 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. The animation on the right shows three different basis elements.

rotating surface

rotaing open set

three basis elements

These animations were created using MATLAB.