This is related to this question.
Let $C = \{(x,y) \in \mathbb{R};\; x^3 + x^2 - y^2 = 0\}$ equipped with the subspace topology of the euclidian plane. I want to show that there's a neigbourhood $U$ of $0 \in C$ which is homeomorphic to a cross $X = (-1,1) \times \{0\} \cup \{0\} \times (-1,1)$.
This comes from an assignment to show that $C$ is no manifold. While visually clear, I want to do this as rigorously as possible.