$\{\mathcal{z}\in\mathbb C| \mathcal{Re(z)} \lt 2\text{ and }\mathcal{Im(z)}\ge 3\}$
Determine whether the set is open, closed, neither or connected and justify your answer.
I am struggling on how to approach this question.
Consider the following subset of $\mathbb{C}$
2
$\begingroup$
general-topology
complex-numbers
2 Answers
0
If I rewrote the problem:
$$\{ (x,y) \in \mathbb R^2 \mid x<2 \, \,, y \geq 3\}$$ would that be helpful?
Recall that the product topology is equivalent to the usual topology in $\mathbb R^n$, so you can ask if $(U \times V)$ is open when $U:=\{x \in \mathbb R \mid x<2\}$ and $V:=\{y \in \mathbb R^2 \mid y \geq 3\}$.
You should also notice that this space must be connected (since it is convex as was stated in another answer.
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Interpret this set as a subset of a plane. A topology in $\Bbb C$ is the ordinary Euclidean topology. This set is not open: find the argument. For closedness please find the answer by yourself. This set is convex, so it is connected.