Show that $I$ and $\mathbb{R}$ are contractible.

A space $X$ is said to be contractible if the identity map $i_X:X\to X$ is nulhomotopic.

If $i_I$ is the identity function of $I:=[0,1]$ in $I$, could I consider the homotopy $H:I\times I\to I$ such that $H(x,t)=tx$, and so the identity function would be homotopic to the zero constant function and with this would it be ready? Does the same function serve to show that $\mathbb{R}$ is contractible? Thank you very much.

  • $\begingroup$ $H(x,1)$ is a constant map? $\endgroup$ – mucciolo Dec 14 '17 at 3:22
  • $\begingroup$ At t = 0. @mucciolo.. $\endgroup$ – William Elliot Dec 14 '17 at 3:38
  • $\begingroup$ No a differnt function is needed for R. $\endgroup$ – William Elliot Dec 14 '17 at 3:40
  • $\begingroup$ @WilliamElliot We could have left him thinking more about it :p $\endgroup$ – mucciolo Dec 14 '17 at 3:43
  • $\begingroup$ @WilliamElliot Why doesn't the same function work for $\mathbb{R}$? $\endgroup$ – KLG Oct 12 '18 at 4:06

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