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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 $\\iota^{-1}(U) = \\set{s \\in S : \\iota(s) \\in U}$.\n2. Since $\\iota(s) = s$, this is $\\set{s \\in S : s \\in U} = S \\cap U$.\n3. By the definition of the [[subspace topology|0D5YHQGDJY]], $S \\cap U$ is open in $S$.\n4. So the preimage of each open set of $X$ is open in $S$, and $\\iota$ is [[continuous|0SY9ECQ19Q]].\n\n## Idea\n\nThe subspace topology is built from the sets $S \\cap U$ with $U$ open in $X$. These are exactly the preimages of open sets under the inclusion, so the inclusion is continuous by design.\n",
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