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Let $(X,d)$ be a metric space, $x \in X, r \in \mathbb{R}, r>0$.
Let $B_{r}(x)$ be the open ball with center $x$ and radius $r$, and let $A^{c}$ denote the complement of any subset $A\subseteq X$.
Let $y\in B_{r}(x)$.

Question
Is it true that $d(y,(B_{r}(x))^{c})\leq r$?

So far I found that if $B_{r}(y)\not\subseteq B_{r}(x)$, then $\exists z\in (B_{r}(x))^{c}: d(y,z)<r$, and thus $$d(y,(B_{r}(x))^{c})\leq d(y,z)<r$$
Otherwise, $B_{r}(y)\subseteq B_{r}(x)$, so $(B_{r}(y))^{c}\supseteq (B_{r}(x))^{c}$, and thus $$r\geq d(y,(B_{r}(y))^{c})\leq d(y,(B_{r}(x))^{c})$$ And here I am stucked.

Does anyone have some ideas? Thanks in advance for helping.

Mizi
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1 Answers1

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No, not necessarily. For instance, if $X=\{0,1\}$ with $d(x,y)=\lvert x-y\rvert$, then $d(0,(B_{1/2}(0))^c)=1$.