Are open disks defined on the unit sphere $S^2$ homeomorphic to open disks on $\mathbb{R}^2$? I know that the unit sphere is a 2D manifold, but that tells me the (seemingly weaker?) point that any point on the sphere has an open neighbourhood that is homeomorphic to a Euclidean open disk. Can we force that neighbourhood to be an open disk on $S^2$?
Context: I am trying to understand a proof that $\mathbb{R}\mathbb{P}^2$, defined as the quotient of a unit sphere by antipodal images, is a manifold. It involves taking a closed disk around a point $p \in S^2$, noting that $S^2$ is compact, and so the quotient map restricted to this disk is actually a homeomorphism. Then we restrict it to the open disk that is the interior.