# Deformation retract

of a topological space $X$
A subset $A\subset X$ with the following property: There exists a homotopy of the identity mapping of $X \to X$ to some mapping $X\to A$ under which all points of the set $A$ remain fixed. If, under the homotopy, the points from $X\setminus A$ remain in $X\setminus A$, $A$ is known as a strong deformation retract. A deformation retract of a space $X$ has the same homotopy type as does $X$. See also Retract; Retract of a topological space.