Homeomorphisms of the cuspidal cubic

Question

Are the cuspidal cubics $V(y^2-x^3)\subset \mathbb{A}^2$ and $V(X^3-Y^2Z)\subset \mathbb{P}^2$ homeomorphic to $\mathbb{A}^1$ and $\mathbb{P}^1$? I think I can see the homeomorphism in my mind (just "pull" from the sides of the cusp) but would like little more rigorus explanation.

Answer 1

If you're asking whether they are isomorphic to $\mathbf A^1$ and $\mathbf P^1$ respectively (which is the right question to ask in this context - the underlying topology is not so interesting, as Alex points out), then the answer is no. Indeed, the cuspidal cubic has a singularity, while $\mathbf A^1$ and $\mathbf P^1$ are smooth.