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  "body": "\n## Definition\n\nLet $X$ and $Y$ be topological spaces.\nA continuous map $f \\colon X \\to Y$ is *null-homotopic* if it is [[homotopic|F2F143KQ2M]] to a constant map $X \\to Y$.\n\n## Remarks\n\nExample: every continuous map $f \\colon X \\to \\R^n$ is null-homotopic, via $H(x, t) = (1 - t) f(x)$.\nNon-example: the identity of the circle $S^1$ is not null-homotopic, since a constant map induces the trivial map on the [[fundamental group|S1XK3HXHFX]] while the identity induces the identity of $\\Z$.\nA map $f \\colon S^n \\to Y$ is null-homotopic if and only if it extends to a continuous map on the closed ball $D^{n+1}$.\nA space $X$ is [[contractible|R5WP8Z69Q8]] if and only if it is nonempty and $\\id_X$ is null-homotopic.\n",
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