Question

Which of the following subsets of C are open and which are closed?
1) {z : |z|<1}
2) the real axis
3)\(\{z:z^n =1~~for~~some~~integer~~n\geq 1\} \)
4)\(\{z\in \mathbb C: z ~~is~~real~~and~~0\leq z<1\}\)
5) \(\{z\in \mathbb C: z~~is~~real~~and~~0\leq z\leq 1\}\)
Please, help

Answer 1

@thomas5267

Answer 2

It is easy to see that 1) is open
2) is open
3) is open

4) is not open nor closed
6) is closed
But how to prove them briefly?

Answer 3

Not sure, not a set theorist.

According to wikipedia, a set is closed if its complement is open. A subset \(U\) of the Euclidean n-space \(\mathbb{R}^n\) is called open if, given any point \(x\in U\), there exists a real number \(\epsilon > 0\) such that, given any point \(y\in \mathbb{R}^n\) whose Euclidean distance from x is smaller than \(\epsilon\), \(y\) also belongs to \(U\). Equivalently, a subset U of \(\mathbb{R}^n\) is open if every point in \(U\) has a neighborhood in \(\mathbb{R}^n\) contained in U.

Answer 4

So I guess for 1:
\[
S_1=\{z : |z|<1\}\\
\forall x\in S_1,|x-y|<|x|-1\implies y\in S_1
\]