I am studying algebraic topology at the moment and we just started with introducing a bunch of definitions that we will use throughout the course. One of those definitions is:
Definition: Let $A\subset B$ be topological spaces, a map $r:B\to A$ is called a retract if $r|_A=id_A$. We say $A$ is a retraction of $B$.
Here $id_A$ stands for the identity on $A$. The definition is clear to me but I am lacking intuition maybe because the only example we have seen is the retract to a single point.
Can someone maybe give some more examples with explanation on what is going on, so that I can understand this concept better intuitively. Thank!
Here is another example: the circle $S^1$ is a retract of $\Bbb R^2\setminus\{(0,0)\}$: consider the map$$\begin{array}{rccc}r\colon&\Bbb R^2\setminus\{(0,0)\}&\longrightarrow&S^1\\&v&\mapsto&\frac1{\|v\|}v.\end{array}$$The idea is that you can (in a continuous way) make the whole space $B$ collapse into $A$ in such a way that no element of $A$ is affected (that is, each element of $A$ is mapped into itself).