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  "body": "\n## Definition\n\nLet $X$ be a set.\nA topology on $X$ is a set $\\tau$ of [[subsets|0FE8Q5S42Y]] of $X$ with these three properties:\n\n1. The [[empty set|0ECVK59Y03]] $\\varnothing$ and the set $X$ are in $\\tau$.\n2. For each family $(U_i)_{i \\in I}$ of members of $\\tau$, the [[union|05GK32H72H]] $\\bigcup_{i \\in I} U_i$ is in $\\tau$.\n3. For all $U, V \\in \\tau$, the [[intersection|0HBRW4E6G1]] $U \\cap V$ is in $\\tau$.\n\nA topological space is a pair $(X, \\tau)$ where $X$ is a set and $\\tau$ is a topology on $X$.\n\n## Remarks\n\nThe members of $\\tau$ are the [[open sets|01ZQQDCERP]] of the space. Their complements are the [[closed sets|0G5AV9Q4RR]].\nBy induction, property 3 gives that each finite intersection of open sets is open. Infinite intersections need not be open.\nEach metric gives a topology, the [[metric topology|0A6YRNZB2R]]. Other basic examples are the [[discrete topology|0S2TEHX4E1]] and the [[indiscrete topology|0XJ02JBD6S]].\nOne can also define a topology by its closed sets, or by the [[neighbourhoods of its points|09CJ00H3NW]].\n\n## Explanations\n\n### L1 human\n\nA topology tells you which subsets of $X$ count as open, without any distances.\nThe rules copy the properties of open sets in metric spaces. Any union of open sets is open, and a finite intersection of open sets is open.\n",
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