Question

Which of the following sets are closed in C. (Justify your answer)

a) A={z∈C:Rez<1}∪{z∈C:Imz≥1}

b) B={z∈C:-∞<x≤3}

Answer 1

a) Let $z_n=1-1/n\in A$, since for all $n\in\mathbb{N}$ $Re\,z_n<1$.

$a=\lim\limits_{n\to\infty}z_n=1\notin A$, because $Re\, a=1$ and $Im\, a=0<1$. Therefore, the set $A$ is not closed.

b) Let $z_n$ be an arbitrary sequence in $B$ converging to a limit $b=\lim\limits_{n\to\infty}z_n$.

$|Re\, b-Re\, z_n|=|Re\, (b-z_n)|\leq |b-z_n|\to 0$. Then $Re\, b=\lim\limits_{n\to\infty}Re\, z_n\leq 3$.

So, $b\in B$ and therefore, $B$ is a closed set.