There's a part of the definition of deformation retraction that I keep forgetting, and that is because I cannot understand why it is required.
The definition states (from Hatcher's Algebraic Toplogy):
A deformation retraction of a space $X$ onto a subspace $A$ is a family of maps $f_t \colon X \to X$, $t \in I$, such that $f_0 = \mathrm{Id}$, $f_1(X) = A$, and $f_t|A = \mathrm{Id}$ for all $t$. The family $f_t$ should be continuous.
Why do we need the requirement $f_t|A = \mathrm{Id}$ for all $t$? Where might things break if this is not required? Is it because we want $A$'s homology to not be potentially modified along the way? If there's an example where this matters that would be great (I played around with $D^2$ but couldn't really identify a potential issue beyond the homology intuition I just mentioned).