My question relates to the following problem and its solution, as described in the comments section of the post.
Specifically, in finding an atlas for the given ellipsoid, the solution proposed comprises the charts given by the 'inverse map' of $(x,y) \to \left(x,y,c\sqrt{1-x^2/a^2-y^2/b^2}\right)$. Is the image of the inverse of this map really an open set? It doesn't look like one.