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

Printable View

- Aug 27th 2011, 04:23 PMrqeeb2 disjoint clsed subsets of (R^2)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

- Aug 27th 2011, 06:31 PMhatsoffRe: 2 disjoint clsed subsets of (R^2)
No, it is not correct. Notice that $\displaystyle \{1/n:n\in\mathbb{Z}^+\}=\mathbb{R}^+$. So $\displaystyle A=\{(r,0):r\in\mathbb{R}^+\}$, and this set is not closed since $\displaystyle (0,0)$ is a boundary point not in $\displaystyle A$.

However, consider the sets $\displaystyle C=\{(x,1/x):x\in\mathbb{R}^+\}$ and $\displaystyle D=\{(x,0):x\in\mathbb{R}\}$. - Aug 27th 2011, 09:35 PMrqeebRe: 2 disjoint clsed subsets of (R^2)
- Aug 28th 2011, 04:09 AMhatsoffRe: 2 disjoint clsed subsets of (R^2)
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.