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

|dw:1441998460778:dw|

You define \(\varepsilon = |x|-1\), right?

ok, thanks for being here. :)

3 also doesn't seems to be open. Its complement should be open with \(\epsilon=\min\{x:k\in \mathbb{Z}^+\land 0\leq k

What is B(x,imaginarypart/2)?

The ball with center x and radius = imaginary part of x /2

if this ball belongs to the set, then the set is open

|dw:1442000196119:dw|

5 is closed by a similar construction.

Thank you so much. :)

just one small thing: the epsilon you define always /2 to have radius of the ball.

because the point is a center. |dw:1442000626403:dw|

:)

But \(|x-y|\color{red}{<}\epsilon\implies y\in U\).