Definition 9.2 (Retraction):

Given a topological space $X$, a subset $A \subseteq X$ is said to be retract of $X$ if there exits a continuous function $r: X \longrightarrow A$ such that $r(a)=a$ for all $a \in A$.

It is immediate that a retract of a Hausdorff space must be closed. The condition that $A$ be a retract of $X$ is quite a strong condition. For example if $X$ is compact and connected then so must $A$. Thus $\{0,1\}$ cannot be a retract of $[0, 1]$. The boundary $\dot{I}^2$ of $I^2$ is not a retract of $I^2$ but this is highly non-trivial.