I am trying to see if the statement is correct. I think it should be.
An irreducible analytic sub-variety $V$ of a complex manifold $M$ is connected. The locus of smooth points of $V$ is denoted as $V^*$ and the locus of the singular points are denoted as $V_{\text{s}}$. Now I want to use the result: An analytic variety $V$ is irreducible iff $V^*$ is connected. Now clearly $V^*$ is open. I want to say that $V^*$ is also dense. Does it follow from Sard's theorem? because locally $V$ looks like zeroes of a finite collection of holomorphic functions. And then I want to say that $V^*$ is connected $\Rightarrow \overline{V^*}=V$ is connected. Can anyone help?