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I've just started with complex analysis and have some questions about closed and open sets in the set of complex numbers. I would be glad if somebody could help me or give me a little hint how to move on.

The exercise is to determine of the following sets $M$, $N$ and $K$ are open/closed and to find their closure, interior and boundary.

1) $M=\left \{ z=x+iy \mid x\leq 1, y\leq 1 \right \}$

Here i think the set is closed, its closure is the set itself, interior is $M=\left \{ z=x+iy \mid x<1, y<1\right \}$ and the boundary $M=\left \{ z=x+iy\mid x= 1, y=1 \right \}$.

2) $N=\left \{ z\in \mathbb{C}\mid \lvert z\rvert \leq 1, Im(z)>0 \right \}$

Here i think this must be the closed half circle in the complex plane. The closure is the set itself, the interior is the open half circle $N=\left \{ z\in \mathbb{C}\mid \lvert z\rvert < 1, Im(z)>0 \right \}$, the boundary $N=\left \{ z\in \mathbb{C}\mid \lvert z\rvert =1, Im(z)>0 \right \}$.

3) $K=\left \{ z\in \mathbb{R} \mid a <z<b \right \}$
Here i am not sure at all...i've been thinking that i can consider $z$ as dense in $\mathbb{R}$, but it's not said that $z$ can be a rational number...

Thank you in advance!

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  • $\begingroup$ I thought this is to ask if there is any closed and open sets in $C$... $\endgroup$ – awllower Feb 20 '13 at 13:59
  • $\begingroup$ 1) is not correct. The boundary is wrong. (Draw a picture, you will see that the boundary consists of two rays, not just a point.) $\endgroup$ – mrf Feb 20 '13 at 14:08
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1) The boundary is the union of the rays $\{1+iy, y\le1\}\cup\{x+i:x\le1\}$

You said the border was a single point.

2) The set is neither. You need to have $Im(z)\ge0$ for it to be closed. The boundary is the set you pointed united with a interval of the real line $[-1,1]$.

3) The set is neither. It is an interval on the real line. Its boundary and its closure are the set itself with the endpoints added. Its interior is empty.

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  • $\begingroup$ Actually, 1) is not correct. (The boundary is wrong.) $\endgroup$ – mrf Feb 20 '13 at 14:04
  • $\begingroup$ Have one more go at it, this time, you read it the same way as I did at first. $\endgroup$ – mrf Feb 20 '13 at 14:13
  • $\begingroup$ Thank you for the corrections, @HenningMakholm. $\endgroup$ – superAnnoyingUser Feb 20 '13 at 14:24
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    $\begingroup$ Now it looks right, except for the "$z=$" which doesn't make any sense in this context. I'd just write it as $\{1+iy\mid y\le 1\}\cup\{x+i\mid x\le 1\}$, though. $\endgroup$ – hmakholm left over Monica Feb 20 '13 at 14:27
  • $\begingroup$ thank you all for the help! I have a question about the last set: can't we consider the last set as an annulus or this will be wrong? $\endgroup$ – Lullaby Feb 20 '13 at 16:01

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