Disclaimer: I have not yet learned about the higher homotopy groups, so any answer not using them would be best for me. The following problem occurred on a recent topology qualifier from Rice University: Let $X$ be a topological space, $A \subset X$ a subspace, and $r: X \to A$ a retraction (i.e...

Disclaimer: I have not yet learned about the higher homotopy groups, so any answer not using them would be best for me. The following problem occurred on a recent topology qualifier from Rice University: Let $X$ be a topological space, $A \subset X$ a subspace, and $r: X \to A$ a retraction (i.e...