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  "body": "\n## Statement\n\nLet $(X_i)_{i \\in I}$ be a family of [[topological spaces|0CNQXVBHCX]], let $X = \\prod_{i \\in I} X_i$ have the [[product topology|04S9HX3GAP]], and let $\\pi_i \\colon X \\to X_i$ be the projections.\nLet $Z$ be a topological space and let $f \\colon Z \\to X$ be a function.\nThen $f$ is [[continuous|0SY9ECQ19Q]] if and only if the [[composition|07HF9YCWPP]] $\\pi_i \\circ f \\colon Z \\to X_i$ is continuous for each $i \\in I$.\n\n## Remarks\n\nProof of the converse: preimages of the subbasic sets $\\pi_i^{-1}(U)$ under $f$ are $(\\pi_i \\circ f)^{-1}(U)$, which are open. Preimages respect unions and intersections, so all preimages of open sets are open.\nThus a map into a product is continuous if and only if all its coordinate functions are continuous. For example, $t \\mapsto (\\cos t, \\sin t)$ is continuous from $\\mathbb{R}$ to $\\mathbb{R}^2$.\nThe statement fails for the [[box topology|0CTM1EHG9P]].\n\n## Explanations\n\n### L1 human\n\nTo check that a map into a product is continuous, check each coordinate separately.\nThis property characterises the product topology among all topologies on the product.\n",
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