Retraction

Let $X$ be a topological space and $A$ a subspace of $X$. A continuous function $r:X \to A$ such that $r(x)=x$ for all $y \in A$ is called a retraction. We also say that $A$ is a retract of $X$.

Another way of looking at this is that $$ri_A=Id_A,$$ where $i_A$ is the inclusion of $A$ into X and $Id_A$ is the identity map on $A$. This equation helps to compare the homology groups of $A$ and $X$: $$r_*i_{A*}=Id_{H_*(A)}.$$