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?
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1A heuristic way to think about this is that every manifold of dimension $n$ has a tangent space of dim $n$ at every point. Now ask, what is the tangent space at $(0,0)$. – Xuqiang QIN Nov 17 '16 at 21:08
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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.
