{
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  "body": "\n## Statement\n\nLet $D \\subseteq \\mathbb{C}$ be a [[domain|097KNJFJ7V]] and let $f : D \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.\nAssume that $f$ is not constant on $D$.\nThen $f$ is an open map: for each open set $V \\subseteq D$, the image $f(V)$ is an open subset of $\\mathbb{C}$.\n\n## Remarks\n\nThe [[maximum modulus principle|048F2P90WR]] follows. If $|f|$ had a local maximum at $z_0$, then $f$ would map a small disc around $z_0$ onto a set that contains no disc around $f(z_0)$.\nThe connectedness of $D$ matters only to exclude functions that are constant on one part of $D$.\nThere is no real analogue: $x \\mapsto x^2$ maps $(-1, 1)$ onto $[0, 1)$, which is not open in $\\mathbb{R}$. In contrast, $z \\mapsto z^2$ maps the unit disc onto the unit disc.\n\n## Explanations\n\n### L1 human\n\nA nonconstant holomorphic function never squeezes an open set into a curve or a point.\nAround each point of the image, the image also contains a small disc.\n",
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