Show that $xy=0$ is a variety but not a manifold

Question

As a subset of $\mathbb{R}^2$ the equation $xy=0$ is the union of $x=0$ and $y=0$. That is two lines. This is an algebraic variety. Why is this not a manifold?

Answer 1

At all points other than that intersection, this variety is locally homeomorphic to $\mathbb R$. If it is a manifold, it should therefore by locally homeomorphic to $\mathbb R$ at every point. At the intersection, deletion of a single point splits the neighborhood into four connected components. But deletion of a point in $\mathbb R$ splits the neighborhood of that point into only two connected components. So they are not locally homeomorphic.