Let $(X,d)$ be any metric space. And $A\subseteq X$ is a bounded set.
Does it mean $\partial A$, the boundary of $A$, is also bounded? It seems it will be bounded. But to prove that, I am confused how to approach it. Any hint is welcome.
2026-03-30 03:35:28.1774841728
Does any bounded set $A$ have bounded boundary $\partial A$?
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No, it doesn't mean that. For instance, in $\mathbb R$ (with the usual metric), $\mathbb{R}\setminus\{0\}$ is unbounded, but its boundary is $\{0\}$, which is bounded.
However, it is true that the boundary of a bounded set $A$ is always bounded. That's because $A\subset B(a,r)$ for some $a\in X$ and some $r>0$, and therefore the boundary of $A$ is contained in the closed ball centered at $a$ with radius $r$.