metric spaces

• Jan 25th 2009, 06:13 PM
dori1123
metric spaces
Define a metric $\displaystyle d$ on a set $\displaystyle X$ having more than one element as follows:
For every $\displaystyle x,y \in X$, d(x,y)={1 if x =/= y, 0 if x = y}
Let $\displaystyle b$ in $\displaystyle X$. Give an example that shows the closure of B(b,1) =/= {y in Y : d(b,y)<=1}
NOTE: B(b,1) is an open ball centered at b with radius 1.
• Jan 25th 2009, 07:14 PM
ThePerfectHacker
Quote:

Originally Posted by dori1123
Define a metric $\displaystyle d$ on a set $\displaystyle X$ having more than one element as follows:
For every $\displaystyle x,y \in X$, d(x,y)={1 if x =/= y, 0 if x = y}
Let $\displaystyle b$ in $\displaystyle X$. Give an example that shows the closure of B(b,1) =/= {y in Y : d(b,y)<=1}
NOTE: B(b,1) is an open ball centered at b with radius 1.

By definition $\displaystyle B(b,1) = \{ x\in X | d(x,b) < 1 \} = X - \{ b \}$.
Now $\displaystyle B\left(b,\tfrac{1}{2}\right) = \{b\}$ does not intersect $\displaystyle X - \{ b\}$.
Thus, $\displaystyle \{b\}$ is not a boundary point.
Thus, $\displaystyle \overline{ B(b,1)} = X - \{b\}$.
Now, $\displaystyle \{x\in X: d(x,b)\leq 1\} = X$.

And we see that $\displaystyle X\not = X-\{b\}$.
• Jan 26th 2009, 12:26 AM
Moo
Hello,