Q: $n$-dimensional complex vectors form a connected set Consider the following set
$$X = \left\{x \in \mathbb{C}^n \; \vert \; \vert Re(x) \vert > \vert Im(x) \vert\right\}$$
where $n\geq 3$ and $Re(x)$ and $Im(x)$ are the real and imaginary parts of $x$ respectively. I want to know if this set is connected or not.
I have looked at the case where $n=1$ (I know $n\geq 3$ in the statement above but I wanted to do it as an exercise). For this, I think that the set is disconnected because if I define the two sets
$$Y = \left\{x \in X \; \vert \; Re(x) > 0\right\}$$
$$Z = \left\{x \in X \; \vert \; Re(x) < 0\right\}$$
then I have two disjoint open sets whose union is $X$ and therefore, $X$ must be disconnected.

*

*Do people agree with my logic here for the case where $n=1$?


*Also, does anyone have any suggestions on how I might approach the $n\geq 3$ case.
