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  "body": "\n## Definition\n\nLet $(X_i)_{i \\in I}$ be a family of [[topological spaces|0CNQXVBHCX]] and let $X = \\prod_{i \\in I} X_i$ be the [[Cartesian product|0C81YBVTWH]].\nThe box topology on $X$ is the topology with [[basis|06EX0FHBTF]] the set of all products $\\prod_{i \\in I} U_i$, where each $U_i$ is open in $X_i$.\n\n## Remarks\n\nFor finitely many factors, the box topology is the [[product topology|04S9HX3GAP]].\nFor infinite products, the box topology is finer than the product topology, and often strictly finer.\nOn $\\mathbb{R}^{\\mathbb{N}}$ with the box topology, the map $t \\mapsto (t, t, t, \\ldots)$ from $\\mathbb{R}$ is not continuous, although each coordinate is continuous. The preimage of $\\prod_{n \\ge 1} (-1/n, 1/n)$ is $\\{0\\}$, which is not open.\n\n## Explanations\n\n### L1 human\n\nThe box topology lets you restrict all coordinates at the same time.\nThis looks natural, but it gives too many open sets, and simple maps into the product stop being continuous.\n",
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