Open ball

For a metric space [math](X,d)[/math] an "open ball" of radius [math]r[/math] centred at [math]a[/math] is the set [math]\{x\in X|d(a,x)\lt r\}[/math], it can be denoted several ways. I frequently encounter

[math]B_r(a)=B(a;r)=\{x\in X|d(a,x)\lt r\}[/math] and use [math]B_r(a)[/math]

Proof that an open ball is open

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TODO: The proof is really easy, just show a smaller ball fits inside, thus the open ball is a neighborhood to all of its points

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See Also

• Open set