I'm studying a real algebraic variety $S$ (embedded in a real Euclidean space) that is "the same at every point" in the following sense: for any two points $p, q \in S$, there is a homeomorphism from $S$ to $S$ that maps $p$ to $q$. (If it helps, that homeomorphism is just an invertible linear transformation in the ambient Euclidean space.)
Does this imply that $S$ is a manifold? If so, where should I look for the math to verify it? If not, can you think of a counterexample?
An obvious implication of this property is that either every point has an open neighborhood homeomorphic to a ball (hence $S$ is a manifold) or no point has an open neighborhood homeomorphic to a ball. So a closely related question is: does there exist a real algebraic variety in which no point has an open neighborhood homeomorphic to a ball? (If "no", then the answer to my first question is "yes".)
BTW, I need an answer to the first question for a research paper I'm writing.