# Characterisation of closed balls in terms of diameters

True or false: The closed balls of a metric space are precisely those subsets such that every proper superset has strictly greater diameter.

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you mean smaller? –  mathemagician Jan 19 '13 at 15:28
False: take $X=\{(-2,0),(0,0),(2,0),(0,3)\}$ in $\mathbb R^2$ with the induced metric. Then the closed ball centered at $(0,0)$ of radius $2$ is the set $B=X\setminus\{(0,3)\}$, and we have $\mathrm{diam}\ X=\mathrm{diam}\ B=4$.