Retract spaces- Material or book recomendations Good afternoon to all!
I have just done a course on Algebraic Topology and I came across the definition of a retract space. Not much more is mentioned with regards to the topological retraction property in the notes I was using. I wonder what the best material to understand this concept more in depth is?
Keeping in mind that I do no have an extensive knowledge on Algebraic Topology, but just the knowledge given by an introductory course on the subject.
The definition of such space which I am mentioning here that is presented in the notes is the following:
Let $Y \subset X$ .  A map $ f: X \rightarrow Y$ is called a retraction of $X$ onto $Y$ if $f$ restricted to $Y$ is the identity map on $Y$. Then $Y$ is called a $retract$ of $X$ .
Any suggestions would be appreciated!
 A: The 'Theory of Retracts' that I referenced in the books above is a lot more involved than this (which is why I doubted your professor would introduce it) . Past a little experience, the definition you give is about as in depth as you will need to understand retracts. 
Spelling out your definition, let $i:A\hookrightarrow X$ be a subspace inclusion. Then $A$ is a retract of $X$ if there is a map $r:X\rightarrow A$, called a retraction, such that
$$r\circ i=id_A.$$
The map $r$ need not be unique. Let $X$ be nonempty. Then the empty set $\emptyset$ is not a retract of $X$. If $x\in X$ is any point, then $\{x\}$ is a retract of $X$. Not every subspace $A\subseteq X$ is a retract. For instance $S^n\subseteq D^{n+1}$ is not a retract (can you prove this using some basic algebraic topology?). If $Y$ is also nonempty, then $X$ is a retract of $X\times Y$ and $X\sqcup Y$. If $A\subseteq X$ is a retract and $X$ is Hausdorff, then $A$ is closed in $X$ (this is an exercise in point-set topology).
There is not much to them theoretically, and they tend to be more useful when recognised in particular examples. The equation $r\circ i=id_A$, through the eyes of an algebraic topologist, is what contains their real power (see the examples above, for instance).
