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  "body": "\n## Statement\n\nLet $(X_i)_{i \\in I}$ be a family of [[compact spaces|0QPHW703PV]], indexed by any set $I$.\nThen the product $\\prod_{i \\in I} X_i$, with the [[product topology|04S9HX3GAP]], is compact.\n\n## Remarks\n\nFor finitely many factors, the statement follows from [[product of compact sets is compact|09ZKE5XQ74]].\nFor arbitrary products, the theorem is equivalent to the axiom of choice.\nThe statement fails for the [[box topology|0CTM1EHG9P]]: the box topology on $\\{0, 1\\}^{\\mathbb{N}}$ is discrete and infinite.\nFor example, $[0, 1]^I$ is compact for each set $I$.\n\n## History\n\nAndrey Tikhonov proved the theorem in 1930 for powers of the closed unit interval. He stated the general theorem in 1935, with the remark that the same proof works.\n\n## Explanations\n\n### L1 human\n\nAny product of compact spaces is compact, even with infinitely many factors.\nIt works because a basic open set of the product restricts only finitely many coordinates.\n",
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