Construct a discrete subset such that every point of $\partial D$ is a limit point of discrete subset

Question

1. $D$ is a domain in $\Bbb C$, please construct a discrete subset $E$ of $D$ such that every point of $\partial D\subset \Bbb C$ is a limit point of $E$

2. $D$ is a unbound domain in $\Bbb C$, please construct a discrete subset $E$ of $D$ such that every point of $\partial D$( $\infty \in \partial D$ is seen as subset of the extended plane) is a limit point of $E$

   Definition: $E$ is a discrete subset of $D$ if $E$ has no limit point that belongs to $D$

---

the problem is related to The Weierstrass factorization theorem

Well, if $D=\Bbb C$, let $ E$ be the positive integers. $\partial \Bbb C=\{\infty\}$

For example , What about $D$ if $D$ is the unit disk $\{z:|z|<1\}$ ?

Answer 1

In this note you'll find a proof that in a metric space, the boundary of any open set is the set of limit points of a discrete subset, which is what you need.