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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]] and give $X \\times Y$ the [[product topology|0Z57EC6YX3]].\nThen the projections $\\pi_1 \\colon X \\times Y \\to X$, $(x, y) \\mapsto x$, and $\\pi_2 \\colon X \\times Y \\to Y$, $(x, y) \\mapsto y$, are [[continuous|0SY9ECQ19Q]].\n\n## Remarks\n\nProof: for $U$ open in $X$, we have $\\pi_1^{-1}(U) = U \\times Y$, which is a basic open set. The proof for $\\pi_2$ is the same.\nThe projections are also [[open maps|0SQ2PC596Y]], but in general they are not closed maps.\nThe product topology is the coarsest topology on $X \\times Y$ that makes both projections continuous.\n\n## Explanations\n\n### L1 human\n\nTaking one coordinate of a point is a continuous operation.\nFor example, the map $\\mathbb{R}^2 \\to \\mathbb{R}$, $(x, y) \\mapsto x$, is continuous.\n",
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