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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nTwo points $x, y \\in X$ are topologically indistinguishable when every [[open set|02BH3MCBQB]] of $X$ that contains one of them also contains the other.\nThis is an [[equivalence relation|0G8VEZ3WTK]] $\\equiv$ on $X$.\nThe *Kolmogorov quotient* of $X$ is the quotient set $\\mathrm{KQ}(X) = X / {\\equiv}$ with the [[quotient topology|0NQ1H2X495]].\n\n## Remarks\n\nEquivalently, $x \\equiv y$ if and only if $\\overline{\\set{x}} = \\overline{\\set{y}}$.\nThe space $\\mathrm{KQ}(X)$ is a [[T0 space|07SG12TQH3]].\nThe quotient map $q \\colon X \\to \\mathrm{KQ}(X)$ is open and closed, and $U \\mapsto q(U)$ is a bijection from the open sets of $X$ to the open sets of $\\mathrm{KQ}(X)$.\nIf $X$ is already $T_0$, then $q$ is a homeomorphism.\n\nExample: let $(X, d)$ be a [[pseudometric space|0A3HJX3KTF]] with its topology.\nHere $x \\equiv y$ if and only if $d(x, y) = 0$, because a point with $d(x, y) \u003e 0$ has a ball around $x$ that misses $y$.\nSo $\\mathrm{KQ}(X)$ is the metric space obtained by identifying points at distance $0$.\n\nNon-example: the identity map of a two-point set $\\set{a, b}$ with the trivial topology is not a Kolmogorov quotient map.\nIts target is not $T_0$, since $a$ and $b$ have the same neighbourhoods; the Kolmogorov quotient of this space is a single point.\n",
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