Deformation retract

Summary

A subspace $A$ of a space $X$ is a deformation retract when $X$ can be deformed continuously onto $A$. At the end of the deformation, the points of $A$ are back where they started. Then the inclusion of $A$ in $X$ is a homotopy equivalence, so $A$ and $X$ have the same homotopy invariants. Deformation retractions are the main tool for replacing a space by a simpler one of the same homotopy type.