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  "body": "\n## Definition\n\nLet $(X_i)_{i \\in I}$ be a family of [[topological spaces|0CNQXVBHCX]], indexed by any set $I$, and let $X = \\prod_{i \\in I} X_i$ be the [[Cartesian product|0C81YBVTWH]], with projections $\\pi_i \\colon X \\to X_i$.\nThe product topology on $X$ is the topology generated by the [[subbasis|0CHR35YNJQ]]\n$$\\{ \\pi_i^{-1}(U) : i \\in I, \\ U \\text{ open in } X_i \\}.$$\n\n## Remarks\n\nA basis consists of the sets $\\prod_{i \\in I} U_i$, where each $U_i$ is open in $X_i$ and $U_i = X_i$ for all but finitely many $i$.\nFor two factors, this is the [[product topology|0Z57EC6YX3]] of $X_1 \\times X_2$.\nIt is the coarsest topology on $X$ for which each projection $\\pi_i$ is continuous.\nA sequence converges in $X$ if and only if each of its coordinates converges. The [[box topology|0CTM1EHG9P]] is finer and lacks this property.\n\n## Explanations\n\n### L1 human\n\nIn an infinite product, a basic open set can restrict only finitely many coordinates.\nThis choice makes the [[universal property|0M6W4XNST6]] and [[Tychonoff's theorem|0JTRSJKCFB]] work.\n",
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