understanding a statement in the proof of a retract of a contractible space is contractible.

Question

Here is the proof of the question:

My question is:

Why we can assume a contraction of $X$ to a point which lies in $A$? Can not this point be outside $A$? if so, how we will deal with the situation?

Could anyone explain this point for me please?

Answer 1

If the point is outside of $A$, let's call it $x_0$, but choose $a_0 \in A$ and choose a continuous path $\gamma : I \to X$ from $x_0$ to $a_0$.

The following formula defines a new homotopy $F' : X \times I \to X$ which has the desired property of being a homotopy from the identity map to the constant map at $a_0$: $$F'(x,t) = \begin{cases} F(x,2t) & \quad 0 \le t \le \frac{1}{2} \\ \gamma(2t-1) & \quad \frac{1}{2} \le t \le 1 \end{cases} $$