A space $X$ is contractible if there is a homotopy $G:X\times I\rightarrow X$ with $G_0=id_X$ and $G_1=x_0$ the constant map at some point $x_0\in X$. The space $X$ is said to be pointed-contractible (with respect to $x_0$) if $G$ can be chosen such that $G_t(x_0)=x_0$ for all $t\in I$.
An interesting example comes from the comb space, which, as parametrised here, is contractible, and is pointed contractible with respect to the point $(0,0)$, but is not pointed contractible with respect to the point $(0,1)$.
A student of mine posed me the following question. I didn't know an answer, so I'll pitch it to you guys.
Is there a space which is contractible, but not pointed-contractible with respect to any given basepoint?
Best Answer
See Hatcher chapter 0, exercise 6. It is an adaptation of the pathology of the comb space.