Deformation retract:

A subspace $A$ of $X$ is said to be a deformation retract if there exists a continuous map $r: X \longrightarrow A$ such that $r \circ j =$   id$_{A}$ and $j\circ r \sim$   id$_{X}$ where $j$ denotes the inclusion of $A$ into $X$. In particular, $X$ and $A$ have the same homotopy type.