Not all retractions come from deformation retractions

Question

On page $3$ of Hatcher it says "Not all retractions come from deformation retractions" and gives an example of a retract to a point and a deformation retract to a point. If a deformation retract to a point $F: X \times I \to X$ has $F(x, 0) = f(x) = id_X$ and $F(x, 1) = g(x) = x_0$, then isn't $g(x)$ a retraction that comes from this deformation retract $F(x, t)$ since $g(X) = \{x_0\}$ and $g\restriction_{\{x_0\}} (x) = x_0$, which is a retraction? How come it says that this retraction doesn't come from a deformation retraction to a point?

Answer 1

In the example, it is not true that all $X$ can have a deformation retract onto a point. If you know that such a deformation retract exist, then yes, the retraction $r(x) = a$ for all $x\in X$ come from a deformation retract.

But in general, no. A retraction to a point might not comes from a deformation retract. The space $X$, as suggested in the book, has to be path connected at the very least.

The simplest example is $X = \{a, b\}$ with the discrete topology, and $r(x) = a$ for $x = a, b$. $r$ is a retraction, while this retraction cannot come from a deformation retract: If it does, then there is

$$F : X\times [0,1] \to X$$

so that $F(\cdot, 0) = id_X$ and $ F(\cdot, 1) = r$. But consider $\{b\} \times [0,1]$, which is path connected. Then $F(b, 0) = b$ would imply $F(b, t) = b$ for all $t$. In particular, $F(b, 1)\neq a$. Thus $r$ cannot come from any deformation retract.