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  "body": "\n## Proof\n\nWe compute preimages of open sets.\n\n1. Let $U$ be open in $X$. The [[preimage|0FGCFY1AZX]] is $\\pi_1^{-1}(U) = \\set{(x, y) \\in X \\times Y : x \\in U} = U \\times Y$.\n2. The set $Y$ is open in $Y$ by the axioms of a [[topology|0CNQXVBHCX]]. So $U \\times Y$ is one of the sets $U \\times V$ of the [[product topology|0Z57EC6YX3]] basis.\n3. By the definition of a [[basis|06EX0FHBTF]], its members are open. So $\\pi_1^{-1}(U)$ is open in $X \\times Y$, and $\\pi_1$ is [[continuous|0SY9ECQ19Q]].\n4. Let $V$ be open in $Y$. In the same way, $\\pi_2^{-1}(V) = X \\times V$, and $X$ is open in $X$.\n5. So $X \\times V$ is a basic open set, and $\\pi_2$ is continuous.\n\n## Idea\n\nThe preimage of an open set under a projection is a box whose other side is the whole factor. Such a box is one of the basic open sets of the product topology.\n",
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