Let $n \ge 2$, and $V$ be an affine algebraic set in $\mathbb C^n$ and $W$ be an irreducible affine algebraic set in $\mathbb C^n$, with $V \subsetneq W$ ; then is it true that $W \setminus V$ is connected in the Euclidean topology of $\mathbb C^n $ ? Is it path connected in the Euclidean topology ?

I can see that $W \setminus V$ is connected in the Zariski topology of $\mathbb C^n$, but I can't figure out in the Euclidean topology.

  • $\begingroup$ Sean Lawton's answer here answers your question, but ends up referencing a book of Mumford: mathoverflow.net/questions/62843/…. I think you can also use roy smith's answer, and the fact that $\mathbb C$ minus a finite number of points is path-connected, but I'm not sure. $\endgroup$ – Dap Feb 1 '18 at 16:10
  • $\begingroup$ @Dap : But $W \setminus V$ is not an algebraic set I think ... $\endgroup$ – user Feb 1 '18 at 19:03
  • $\begingroup$ You can adapt my answer to your preceding question to see that $W\setminus V$ is connected in the classical topology. The key point is that, denoting by a bar the closure in $ \mathbb P^n$ of a subset of $\mathbb C^n$ , we have $\bar W \setminus \bar V=W \setminus V$. We can then use the result I called the Connectedness theorem in my preceding answer. $\endgroup$ – Georges Elencwajg Feb 1 '18 at 23:50

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