Hatcher's Algebraic Topology Chapter 0 at Page 3

Question

I have a question from Hatcher's Algebraic Topology Chapter 0 at Page 3: http://www.math.cornell.edu/~hatcher/AT/ATch0.pdf

One could equally well regard a retraction as a map $X\to A$ restricting to the identity on the subspace $A \subset X$ . From a more formal viewpoint a retraction is a map $r : X \to X$ with $r^2 = r$ , since this equation says exactly that $r$ is the identity on its image.

So I am confused: why we can see retraction map as restriction? Should I see the restriction here as restriction on a local chart? Assuming so, where the explicit function $r^2 = r$ origin from?

Thank you~

Answer 1

Think of a function $r: X \rightarrow X$ such that $r^2 = r$. The image of this function is $r(X)$, and applying the function twice doesn't change the output after the first application. That is, $r(r(X)) = r(X)$. In particular, $r$ is the identity on $r(X)$; if we relabel and let $A := r(X)$, then we have the sort of definition initially provided.

With regard to why one would use such a definition: this idea of $r^2 = r$ is a specific example of a more general property (of all sorts of different things) called "idempotence." For an example of idempotence from point-set topology, see the second closure axiom here.