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  "body": "\n## Definition\n\nLet $(X, \\tau_X)$ and $(Y, \\tau_Y)$ be [[topological spaces|0CNQXVBHCX]].\nThe product topology on the [[Cartesian product|0C81YBVTWH]] $X \\times Y$ is the topology with [[basis|06EX0FHBTF]]\n$$\\{ U \\times V : U \\in \\tau_X, \\ V \\in \\tau_Y \\}.$$\n\n## Remarks\n\nThis family is a basis of a topology, because $(U_1 \\times V_1) \\cap (U_2 \\times V_2) = (U_1 \\cap U_2) \\times (V_1 \\cap V_2)$.\nA set $W \\subseteq X \\times Y$ is open if and only if for each $(x, y) \\in W$ there are open $U \\ni x$ and $V \\ni y$ with $U \\times V \\subseteq W$.\nFor metric spaces, it is the metric topology of the [[product metric|044D4AQV8V]]. The product topology on $\\mathbb{R} \\times \\mathbb{R}$ is the usual topology of $\\mathbb{R}^2$.\nThe [[projections are continuous|0PM88Z9W4M]]. For products of any family of spaces, see the [[product topology on an arbitrary product|04S9HX3GAP]].\n\n## Explanations\n\n### L1 human\n\nThe open sets of $X \\times Y$ are built from open rectangles $U \\times V$.\nIn the plane, open rectangles and open discs give the same topology.\n",
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