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  "body": "\n## Definition\n\nThe Riemann sphere, or extended complex plane, is the set $\\widehat{\\mathbb{C}} = \\mathbb{C} \\cup \\{\\infty\\}$, where $\\mathbb{C}$ is the set of [[complex numbers|09FTAET5Z4]] and $\\infty$ is one new point.\nIts topology is as follows.\nA set $V \\subseteq \\widehat{\\mathbb{C}}$ is open if the following two conditions hold. First, $V \\cap \\mathbb{C}$ is open in the [[complex plane|0CTE7A31XQ]]. Second, in the case $\\infty \\in V$, $V$ also contains the set $\\{ z \\in \\mathbb{C} : |z| \u003e R \\}$ for some $R \u003e 0$.\nLet $S^2 = \\{ (x_1, x_2, x_3) \\in \\mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \\}$ and $N = (0, 0, 1)$.\nThe stereographic projection $\\sigma : S^2 \\to \\widehat{\\mathbb{C}}$ is $\\sigma(x_1, x_2, x_3) = \\frac{x_1 + i x_2}{1 - x_3}$ for points other than $N$, and $\\sigma(N) = \\infty$.\nThe map $\\sigma$ is a homeomorphism, so $\\widehat{\\mathbb{C}}$ is a sphere as a topological space.\n\n## Remarks\n\nA sequence $(z_n)$ of complex numbers tends to $\\infty$ in $\\widehat{\\mathbb{C}}$ exactly when $|z_n| \\to \\infty$.\nSome arithmetic extends to $\\infty$: $z + \\infty = \\infty$ for $z \\in \\mathbb{C}$, and $\\frac{z}{0} = \\infty$ and $\\frac{z}{\\infty} = 0$ for $z \\neq 0$, but the expressions $\\infty - \\infty$, $0 \\cdot \\infty$, $\\frac{0}{0}$ and $\\frac{\\infty}{\\infty}$ stay undefined.\nWith the value $\\infty$ at each [[pole|0WPQYW8MTE]], a [[meromorphic function|0NSD1TPHR0]] on an open set $U$ becomes a continuous map from $U$ to $\\widehat{\\mathbb{C}}$.\nThe Riemann sphere is a compact Riemann surface, with the two charts $z$ near points of $\\mathbb{C}$ and $\\frac{1}{z}$ near $\\infty$.\n\n## Explanations\n\n### L1 human\n\nImagine a ball that sits on the complex plane at $0$.\nDraw a line from the top of the ball to a point of the plane; the line meets the ball at one more point.\nPoints far from $0$ go to points near the top, so the top of the ball plays the role of $\\infty$.\n",
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