I'm reading Hatcher and on page 12 he writes:
"... It is less trivial to show that there are path-connected spaces that do not deformation retract onto a point...."
The definition of deformation retract is given as: "A deformation retraction of a space $X$ onto a subspace $A$ is a family of maps $f_t :X→X$, $t \in I$, such that $f_0 = 1 $ (the identity map), $f_1(X) = A$, and $f_t |A = 1$ $\forall t$. The family $f_t$ should be continuous in the sense that the associated map $X×I→X, (x,t) \rightarrow f_t(x)$ is continuous."
Now my question is: how is it possible for a space $X$ to be path-connected but not retractable to a single point? Can one not just pick $x \in X$ and then retract any $y \in X$ along the path between $x$ and $y$?