Null-homotopic map

Remarks

Example: every continuous map $f \colon X \to \R^n$ is null-homotopic, via $H(x, t) = (1 - t) f(x)$. Non-example: the identity of the circle $S^1$ is not null-homotopic, since a constant map induces the trivial map on the fundamental group while the identity induces the identity of $\Z$. A map $f \colon S^n \to Y$ is null-homotopic if and only if it extends to a continuous map on the closed ball $D^{n+1}$. A space $X$ is contractible if and only if it is nonempty and $\id_X$ is null-homotopic.