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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CNQXVBHCX]], let $S \\subseteq X$ have the [[subspace topology|0D5YHQGDJY]] and let $\\iota \\colon S \\to X$ be the inclusion map $\\iota(s) = s$.\nThen $\\iota$ is [[continuous|0SY9ECQ19Q]].\n\n## Remarks\n\nProof: for $U$ open in $X$, the [[preimage|0FGCFY1AZX]] is $\\iota^{-1}(U) = S \\cap U$, which is open in $S$.\nThus the [[restriction|0D2MXBV21T]] $f|_S = f \\circ \\iota$ of a continuous map $f \\colon X \\to Y$ is continuous.\nA map $g \\colon Z \\to S$ is continuous if and only if $\\iota \\circ g \\colon Z \\to X$ is continuous. This is the universal property of the subspace topology.\n\n## Explanations\n\n### L1 human\n\nPutting a subspace back into the big space is a continuous map.\nSo when you restrict a continuous function to a subset, you get a continuous function.\n",
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