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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\tau$ be a [[topology|0CQHFPGJ45]] on $X$.\nThe topology $\\tau$ is an *Alexandrov topology* if for every family $(U_i)_{i \\in I}$ of [[open sets|02BH3MCBQB]] of $\\tau$, the intersection $\\bigcap_{i \\in I} U_i$ is open.\nA topological space whose topology is an Alexandrov topology is called an *Alexandrov space* or an Alexandrov-discrete space.\n\n## Remarks\n\nEquivalently, every point $x \\in X$ has a smallest open set $U_x$ containing it, namely the intersection of all open sets that contain $x$.\n\nFor a [[preorder|0VABYWJWH9]] $\\le$ on $X$, the family of all [[upper sets|ETCE287E2R]] of $(X, \\le)$ is an Alexandrov topology.\nConversely, every Alexandrov topology arises in this way from exactly one preorder, its [[specialization preorder|CMYVK9CEKZ]].\n\nExample: every topology on a finite set is an Alexandrov topology, because then every family of open sets is finite.\n\nNon-example: the usual topology of $\\R$ is not an Alexandrov topology.\nThe open sets $(-1/n, 1/n)$ for $n \\ge 1$ have intersection $\\set{0}$, which is not open.\n\nThe name \"Alexandrov space\" is also used in metric geometry for spaces with curvature bounded below; that notion is unrelated.\n",
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