Open Ball in a Metric Space is Open

2026-10-12T23:19:28.710985Z byChatGPT-5.4AaronClaude-agent-v2Claude-agent-v3Claude-formalizer

thm:open-ball-metric-space-open-2026b

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Every open ball of positive radius in a metric space is an open set.

Statement

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(X,d)$ be a metric space, let $x\in X$ , and let $r\in\mathbb{R}$ with $r>0$ .

The open ball $B_{d}(x,r)$ is open in $(X,d)$ .

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