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  "body": "\n## Definition\n\nLet $(X, \\tau)$ be a [[topological space|0CNQXVBHCX]] and let $\\sim$ be an [[equivalence relation|0SGTMBEWRF]] on $X$.\nLet $q \\colon X \\to X / {\\sim}$ be the map to the [[quotient set|0ZVHXDSZC7]] that sends $x$ to its class.\nThe quotient topology on $X / {\\sim}$ is\n$$\\{ V \\subseteq X / {\\sim} : q^{-1}(V) \\in \\tau \\}.$$\n\n## Remarks\n\nIt is the finest topology on $X / {\\sim}$ for which $q$ is continuous.\nA [[preimage|0FGCFY1AZX]] $q^{-1}(V)$ is a union of classes. Thus $V$ is open if and only if the union of the classes in $V$ is open in $X$.\nQuotients need not keep good properties: a quotient of a Hausdorff space need not be [[Hausdorff|0MXB4CKHFW]].\nExamples are the [[circle|0WH1GVBJNH]], the [[torus|03NM7DZXXC]] and the [[Möbius strip|04AJE0M271]].\n\n## Explanations\n\n### L1 human\n\nYou make a quotient space by gluing the points of each class into one point.\nFor example, gluing the two ends of $[0, 1]$ gives a circle.\n",
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