Let D be a domain in $\mathbb{C}$, thus a connected open subset of the complex numbers. Suppose we remove one point of it no matter which one. Is D without this point still connected?
2026-04-25 05:25:48.1777094748
Connected open subset after piercing the set
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Hint In $\mathbb C$ an open set is connected if and only if it is path connected.
Hint 2 Lets call the point you remove $a$. If $P$ is a path from $x$ to $y$ in $D$ passing via $a$, use the fact that there exists some $r$ such that $\{ z: |z-a|< r \} \subset D$ to construct a path from $x$ to $y$ in $D$ not passing through $a$.