deformation retract of

Let X be a space and let A and B be subspaces of X with $A \subset B$. Then A is called a deformation retract of B over X provided that the identity mapping $i_B: B \to B$ is homotopic in X to a retraction $r: B \to A$. Further, A is called a strong deformation retract of B over X provided that it is a deformation retract of B over X and the homotopy keeps the points of A fixed throughout the entire deformation of B into A (see e.g. [41, Definition 6.3, p. 324]).