Sets

A set $S\in \mathbb{C}$ is called an open set if all its points are interior.

An open set $S$ is connected if any $z_1,z_2\in S$ can be joined by a "polygonal line" contained in $S$.

A set $S$ is bounded if $\mathrm{\exists }M>0$: $\mathrm{\forall }z\in S$ $|z|<M$. In other words, $\mathrm{\exists }M$L: $S\subset U_M(0)$