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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nLet $G/H = \\set{gH : g \\in G}$ be the set of [[left cosets|03DEH43F4H]] of $H$, and let $\\pi \\colon G \\to G/H$, $\\pi(g) = gH$.\nThe *quotient* of $G$ by $H$ is the set $G/H$ with the [[quotient topology|0NQ1H2X495]] of $\\pi$.\nSo a set $U \\subseteq G/H$ is open if and only if $\\pi^{-1}(U)$ is open in $G$.\nIf $H$ is a [[normal subgroup|0CB0DX5BVV]], then $G/H$ is also given the structure of the [[quotient group|0KYX2N69J0]].\n\n## Remarks\n\nThe projection $\\pi$ is continuous and an [[open map|0YCG648KDD]]: for $U$ open in $G$, the set $\\pi^{-1}(\\pi(U)) = UH$ is a union of the open sets $Uh$.\nIf $H$ is normal, then $G/H$ is a topological group, and $\\pi$ is a continuous homomorphism.\nThe space $G/H$ is [[Hausdorff|0S1FJ8VEX9]] if and only if $H$ is closed in $G$.\nExample: $\\R/\\Z$ is a compact Hausdorff topological group, isomorphic to the circle group through $x + \\Z \\mapsto e^{2\\pi i x}$.\nNon-example of the Hausdorff property: $\\R/\\Q$ carries the indiscrete topology, because $\\Q$ is dense, so every nonempty open set of $\\R$ meets every coset of $\\Q$.\n",
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