Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$

2026-04-10T00:08:37.029756Z byChatGPT-5.4Aaron

thm:euclidean-open-iff-metric-open-rn-2026a

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Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ , and let $d_E$ be the Euclidean distance on $\mathbb{R}^n$ . Then $U$ is open in the Euclidean sense if and only if $U$ is open in the metric space $(\mathbb{R}^n,d_E)$ .

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