2 disjoint clsed subsets of (R^2)

**rqeeb** · August 27th 2011, 04:23 PM

Q:Find two disjoint clsed subsets of (R^2) which are zero distance a part?

my answer is A & B where

A={(1/n) : n belong to z+}X{0}
B={[-1,0]XR}

Is it right

**hatsoff** · August 27th 2011, 06:31 PM

No, it is not correct. Notice that $\{1/n:n\in\mathbb{Z}^+\}=\mathbb{R}^+$ . So $A=\{(r,0):r\in\mathbb{R}^+\}$ , and this set is not closed since $(0,0)$ is a boundary point not in $A$ .

However, consider the sets $C=\{(x,1/x):x\in\mathbb{R}^+\}$ and $D=\{(x,0):x\in\mathbb{R}\}$ .

**rqeeb** · August 27th 2011, 09:35 PM

Originally Posted by hatsoff

However, consider the sets $C=\{(x,1/x):x\in\mathbb{R}^+\}$ and $D=\{(x,0):x\in\mathbb{R}\}$ .

Thank u for your respond
but $C=\{(x,1/x):x\in\mathbb{R}^+\}$ looks like is not closed since $D=\{(x,0):x\in\mathbb{R}\}$ is boundry point for the set C which not conained in D. Am I right or wrong

**hatsoff** · August 28th 2011, 04:09 AM

D is a set, not a point. So D could not possibly *be* a boundary point.

Think of it graphically: C is just the curve y=1/x in the first quadrant of the plane. D is the line y=0.

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