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  "body": "\n## Description\n\nThe *open unit disc* is $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$, and the *closed unit disc* is $\\overline{\\mathbb{D}} = \\set{z \\in \\C : \\abs{z} \\le 1}$.\nThe open disc is open, since it is the preimage of $(-\\infty, 1)$ under the continuous map $z \\mapsto \\abs{z}$, and it is connected, since it is convex.\nSo it is a [[domain in the complex plane|0EWMDD6C14]].\nIt is star-shaped about $0$, so every closed curve in it contracts to $0$ along straight segments, and it is a [[simply connected domain|0QW64ARSHB]].\nIts boundary is the unit circle $\\mathbb{T} = \\set{z \\in \\C : \\abs{z} = 1}$.\n\n## Remarks\n\nBy the [[Riemann mapping theorem|02WHMEG5ZR]], every simply connected domain other than $\\C$ is conformally equivalent to $\\mathbb{D}$.\nThe Cayley map $z \\mapsto (z - i)/(z + i)$ is a conformal map of the upper half-plane onto $\\mathbb{D}$.\nThe conformal automorphisms of $\\mathbb{D}$ are the [[Möbius transformations|0SG74WBFA2]] $z \\mapsto e^{i\\theta}(z - a)/(1 - \\bar a z)$ with $\\abs{a} \u003c 1$ and real $\\theta$.\nWith the hyperbolic metric it becomes the [[Poincaré disk model|85MM2EA0TK]] of the hyperbolic plane.\n\n## Explanations\n\n### L1 human\n\nThe unit disc is the set of complex numbers at distance less than $1$ from $0$: a round region without its rim.\nIt is the standard domain of complex analysis, because by the Riemann mapping theorem many other regions can be turned into it without distorting small angles.\nFunction spaces such as the [[Hardy spaces|PNNNFJVF6W]] and the [[Bergman spaces|2D619WTPYC]] are usually first defined on the unit disc.\n",
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