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  "body": "\n## Summary\n\nAn Alexandrov topology is a topology in which the intersection of any family of open sets, finite or infinite, is open.\nEquivalently, every point has a smallest open neighbourhood.\nAlexandrov topologies on a set correspond exactly to [[preorders|0VABYWJWH9]] on it: the open sets are the [[upper sets|ETCE287E2R]] of the preorder.\nEvery finite topological space is of this kind, so these spaces are the bridge between finite topology and order theory.\n\n## Explanations\n\n### L1 human\n\nIn most spaces, such as the real line, a point has smaller and smaller neighbourhoods with no smallest one.\nIn an Alexandrov space each point $x$ has a smallest open set around it, and this set says which points are \"above\" $x$.\nSo the whole topology is just a way of writing down an order.\nFor example, on $\\set{1, 2, 3}$ ordered by $1 \\le 2 \\le 3$, the upper sets $\\emptyset$, $\\set{3}$, $\\set{2, 3}$ and $\\set{1, 2, 3}$ form an Alexandrov topology.\n",
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