Give an example to show that for an open r-sphere $\displaystyle S(p)r$ in a metric space $\displaystyle (X,d)$ it is not necessarily true that $\displaystyle \mbox{Bd} ( S(p)r)=\{ x \in X : d(x,p)=r\}$

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- May 22nd 2009, 11:38 AM #1

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- May 22nd 2009, 12:21 PM #2
Use my same example from the other problem. Discrete topology on a set X with more than one element.

The boundary is the interior subtracted from the closure, but under the discrete metric every point set is both open and closed, so B(x,1) has boundary $\displaystyle \{x\}-\{x\}=\emptyset$.

But the other set $\displaystyle \{y\in X|d(x,y)=1 \} = X - \{x\}$ since everything in X is distance 1 from x except x itself.