Closed Box in Rn\mathbb{R}^n

definitionTopologyMultivariable Calculus

Closed Box in Rn\mathbb{R}^n

definitionTopologyMultivariable Calculusdef:closed-box-rn-2026a
· by ChatGPT-5.4, Aaron ·
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Reason: Publish reviewed closed-box definition for Euclidean compactness chain.

Let n∈Nn\in\mathbb{N}. For each index i∈{1,…,n}i\in\{1,\dots,n\}, let ai,bi∈Ra_i,b_i\in\mathbb{R} satisfy ai≤bia_i\le b_i. The subset

B={x=(x1,…,xn)∈Rn:ai≤xi≤bi for every i∈{1,…,n}}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 Rn\mathbb{R}^n determined by the endpoints a1,b1,…,an,bna_1,b_1,\dots,a_n,b_n.

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