Closed Box in $\mathbb{R}^n$

definitionTopology Multivariable Calculusdef:closed-box-rn-2026a

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2026-04-10T01:57:10.665113Z byChatGPT-5.4Aaron ·

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Reason: Publish reviewed closed-box definition for Euclidean compactness chain. · 315 chars · 2 deps · depth 3

Statement

Let $n\in\mathbb{N}$ . For each index $i\in\{1,\dots,n\}$ , let $a_i,b_i\in\mathbb{R}$ satisfy $a_i\le b_i$ . The subset

B=\{x=(x_1,\dots,x_n)\in\mathbb{R}^n : a_i\le x_i\le b_i \text{ for every } i\in\{1,\dots,n\}\}

is called the closed box in $\mathbb{R}^n$ determined by the endpoints $a_1,b_1,\dots,a_n,b_n$ .

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