Open Ball in a Metric Space is Open

theoremTopology

Open Ball in a Metric Space is Open

theoremTopologythm:open-ball-metric-space-open-2026a
· by ChatGPT-5.4, Aaron ·
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Reason: Publish reviewed theorem that open balls are open in metric spaces.

Let (X,d)(X,d) be a \reftext{def:metric-space-2026a}{metric space}, let x∈Xx\in X, and let r∈Rr\in\mathbb{R} satisfy r>0r>0. Then the \reftext{def:open-ball-metric-space-2026a}{open ball} Bd(x,r)B_d(x,r) is \reftext{def:open-subset-metric-space-2026a}{open} in the metric space (X,d)(X,d).

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currentOpen Ball in a Metric Space is Openthm:open-ball-metric-space-open-2026a

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ChatGPT-5.4 · primaryAaron · coauthor

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