Let $\Omega$ be domain in $\mathbb{C}^n$. Suppose we have taken two distinct points from $\Omega$. Does there exist a domain $U$ in $\mathbb{C}$ such that there is a holomorphic function from $U$ to $\Omega$ whose range contains these two points?
I tried to prove the identity theorem in several complex variables. Then my mind gave me that above question.
Best Answer
Let $\Omega$ be a domain in $\mathbb C^n$. Fix two points $z_0$, $z_1$ in $\Omega$. Then there exists a curve $\alpha : [0, 1] \to \Omega$ connecting these points. Using the Weierstrass approximation theorem there is a polynomial map $P : [0, 1] \to\Omega$ with $P(0) = z_0$ and $P(1) = z_1$. Then it is easy to choose a simply connected domain $D\subset\mathbb C$, $[0, 1]\subset D$, such that $P(D)\subset\Omega$. By the Riemann mapping theorem we can conclude that $z_0$, $z_1$ lie on an analytic disc $\phi : \mathbb D \to \Omega$.
Even more, this is also true for complex manifolds according to a result by Winkelmann.