= Deformation retraction
{title2=$H:X\times[0,1]\to X$}
A <deformation retraction> of $X$ onto a subspace $A$ is a <homotopy> $H:X\times[0,1]\to X$ such that $H(x,0)=x$, $H(x,1)=r(x)\in A$, and $H(a,t)=a$ for every $a\in A$ and every $t$. Thus the <retract> is fixed pointwise throughout. With inclusion $i:A\hookrightarrow X$, the equations $r\circ i=\operatorname{id}_A$ and $i\circ r\simeq\operatorname{id}_X$ exhibit a <homotopy equivalence>. See https://pi.math.cornell.edu/~hatcher/AT/ATch0.pdf#page=2[Hatcher, Algebraic Topology, Chapter 0, pp. 2–3].
Take the closed <annulus>
$$
A=\{x\in\mathbb R^2:1\leq\lVert x\rVert\leq2\}.
$$
An exact <deformation retraction> onto its inner <circle> is
$$
H(x,t)=\left((1-t)+\frac{t}{\lVert x\rVert}\right)x,
\qquad r(x)=\frac{x}{\lVert x\rVert}.
$$
The denominator is nonzero everywhere on the <annulus>. If the starting radius is $\rho=\lVert x\rVert$, its radius at time $t$ is $(1-t)\rho+t$. This lies between $1$ and $\rho\leq2$, verifying that the entire <homotopy> stays in the original <annulus>. When $\rho=1$, the point is stationary, as required.
For $x=(0,2)$, times $0,1/2,1$ give $(0,2),(0,3/2),(0,1)$. A radius-$3/2$ <loop (topology)> similarly becomes a radius-$5/4$ <loop (topology)> halfway through, then the unit <circle>. Its angle and <winding number> are preserved. This is useful because computing the <fundamental group> of the <annulus> reduces to computing that of the <circle>; the explicit <homotopy equivalence> establishes why the answer transfers.
\Image[/annulus-deformation-retraction.png]
{title=Radial deformation of the closed annulus onto its fixed inner circle, with intermediate radii staying in the annulus}
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