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I have this set of complex numbers: $\{ 1-i , 2-i , 3-i \}$, and another set $$B:= \{ w \in \Bbb{C} \mid 0 \leq \mathrm{Re}(w)\leq 4 \land -2 < \mathrm{Im}(w)\leq0 \} \setminus \{ a+bi \in \Bbb{C} \mid 2<a<5 \land -1\leq b\leq2\}.$$

I need to check for every $z \in \{ 1-i , 2-i , 3-i \}$ whether the set $B$ is a neighborhood of $z$.

What I did is this:

$B=\{(0-i),(0+0i),(1-i),(1+0i),(2-i),(2+0i)\}$, and now I say: for $z=1-i$,$z=2-i$ the set $B$ can be neighborhood of them, since $z+\epsilon/2$ and $z-\epsilon/2$ still live in $B$.

Can you please correct me? Thanks for any guidance!

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What is your definition for $B$ being a neighbourhood of $z$? (At any rate you should begin by trying to determine what $B$ is as a subset of the complex plane: if you have to, draw it out! It is certainly not a finite set as you indicate in the third paragraph.) –  Arthur Fischer Dec 16 '12 at 18:13
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1 Answer

up vote 2 down vote accepted

It would be helpful to start with a visualization (but pay attention to which portions of the common blue region are open vs closed based on your inequalities given):

Mathematica graphics

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can you please tell me what exactly tells me this visualization –  doniyor Dec 16 '12 at 18:35
    
The bottom rectangle is the first set you specified in the definition of $B$. The top rectangle is the second set you specified. Since $B$ is the "set difference" between these two, think of taking the first rectangle and subtracting off the second. Is the resulting set a neighborhood of each of the three points shown? –  JohnD Dec 16 '12 at 18:44
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