I wonder if my approach is completely wrong. If so, may I request for some hints for heading to the right direction? Thank you!
Show that a retract of a contractible space is contractible.
The previous discussion Proof that retract of contractible space is contractible. used the definition that
The identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.
But as to what I found, a space being contractible is simply defined as
A space having the homotopy type of a point is called contractible.
So following this definition, I can only carried out the proof in such a way:
We retract this space $X$ to $A \subset X$. Since $A$ is a subset of $X$, $A$ is also contractible.
So I am wondering, did I miss the definition somewhere? Or I should just use the other definition?