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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], let $f \\colon X \\to Y$ be [[continuous|0SY9ECQ19Q]] and let $S \\subseteq X$ be [[connected|0R37DFJKFW]].\nThen $f(S)$ is connected.\n\n## Remarks\n\nProof: if open sets $U$ and $V$ split $f(S)$, then the [[preimages|0FGCFY1AZX]] $f^{-1}(U)$ and $f^{-1}(V)$ are open and split $S$.\nThus connectedness is a [[topological property|02SMVGC690]].\nFor $Y = \\mathbb{R}$, this gives the [[intermediate value theorem|098C0KZZAP]].\nFor example, the circle is connected, because it is the image of $[0, 1]$ under $t \\mapsto (\\cos 2\\pi t, \\sin 2\\pi t)$.\n\n## Explanations\n\n### L1 human\n\nA continuous map cannot break a space that is in one piece into separate pieces.\n",
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