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  "body": "\n## Example\n\nLet $S = \\set{0, 1}$ with the family of [[open sets|02BH3MCBQB]] $\\set{\\emptyset, \\set{1}, \\set{0, 1}}$.\nThis family contains $\\emptyset$ and $S$, and it is closed under unions and intersections, as one checks on its three members.\nSo $S$ is a [[topological space|0CQHFPGJ45]], the *Sierpiński space*.\n\n- $S$ is a [[T0 space|07SG12TQH3]]: the open set $\\set{1}$ contains $1$ but not $0$.\n- $S$ is not a [[T1 space|0NW6CH38YR]]: the only open set that contains $0$ is $S$, and it also contains $1$.\n- The [[closed sets|0ZCK8DQYN1]] are $\\emptyset$, $\\set{0}$ and $S$, so the closure of $\\set{1}$ is $S$.\n- $S$ is [[connected|0V032VWTHD]], since its only clopen sets are $\\emptyset$ and $S$.\n\nFor any topological space $X$, a map $f \\colon X \\to S$ is [[continuous|0VGFHTM75Y]] if and only if $f^{-1}(\\set{1})$ is open in $X$.\nSo continuous maps $X \\to S$ correspond one to one to open subsets of $X$.\n\n## Explanations\n\n### L1 human\n\nThe Sierpiński space is the smallest topological space that is neither discrete nor trivial.\nIt has two points, and only one of them, the point $1$, forms an open set by itself.\nOne can read $1$ as \"yes, observed\" and $0$ as \"not yet observed\": an open set of any space is the same as a continuous yes-or-no test on it.\nThis small space is a basic counterexample, since it separates points in one direction only.\n",
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