It seems to me the difference between the pair $(X,A)$ being a good pair and having the HEP is very slight, so this answer is meant as more of a comment to illustrate the differences.
Hatcher (in "Algebraic Topology", just after Theorem 2.13) defines $(X,A)$ to be a good pair if
$X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$.
In the same text, soon after Example 0.14, he defines the pair $(X,A)$ to have the homotopy extension property if, here for only $A$ a subspace of $X$ (no condition on $A$ being closed or non-empty),
$X\times \{0\}\cup A\times I$ is a retract of $X\times I$.
Most other sources do not use the phrase "good pair" and simply stick to HEP. One such source (in my opinion, the best source) is May. There (in "A Concise Course in Algebraic Topology", Chapter 6, Section 1), still only assuming $A$ is a subspace of $X$, the pair $(X,A)$ along with a map (not necessarily the inclusion) $i:A\to X$, is defined to have the homotopy extension property if
$i:A\to X$ is a cofibration.
In Section 4 of the same chapter, now assuming $A$ is closed in $X$, May shows that $(X,A)$ is a neighborhood deformation retract pair if and only if
- $X\times \{0\}\cup A\times I$ is a retract of $X\times I$, or
- the inclusion $i:A\hookrightarrow X$ is a cofibration.
In fact, now it seems Hatcher's definitions of good pair and having the HEP are equivalent from May's viewpoint. That's why, in my opinion, the phrase "good pair" is not the best approach to use, and instead we should talk about cofibrations that are or are not inclusions.
Let's look at Hatcher's definition:
...a homotopy, which is simply any family of maps $f_t : X \to Y , t \in I$ , such that the associated map $F : X × I \to Y$ given by $F(x,t) = f_t(x)$ is continuous. One says that two
maps $f_0,f_1: X \to Y$ are homotopic if there exists a homotopy connecting them, and one writes $f_0 \sim f1$
Thus we Hatcher says that for a map $f_0:X \to Y$ with a subspace $A \subset X$ we are given are homotopy $f_t:A \to Y$ of $f_0|A$ this simply means that there is some other map $f_1:A \to Y$ that is homotopic to $f_0|A$. The homotopy extension property then tells us when this homtopy extends to the whole of $X$ (not just the subspace $A \subset X$)
There are plenty of examples. For example Proposition 0.16 gives a useful one - a CW-pair $(X,A)$ has the homotopy extension property.
This will be especially useful when you learn about cofibrations.
the following picture.
Best Answer
It's definitely doable. Let's consider a simpler example first: let $X=[0,1]$, and let $A=\{0\}$.
You can retract $X\times I$ (a square) to $(X\times\{0\})\cup(A\times I)$ (the union of the "bottom" and "left" sides of the square) by projecting each point along the ray from $(2,2)$:
To move this intuition to your example of $X=$ a disk and $A=$ a smaller disk inside $X$, just "swing this around" (as one would to form a solid of revolution) and leave the interior of $A$ alone.
For fun:
PlotACylinder[RadiusOfA_, Height_, theta_, u_] := {RadiusOfA*Cos[theta], RadiusOfA*Sin[theta], Height*u} PlotATop[RadiusOfA_, Height_, theta_, u_] := {RadiusOfA*u*Cos[theta], RadiusOfA*u*Sin[theta], Height} PlotX[RadiusOfX_, theta_, u_] := {RadiusOfX*u*Cos[theta], RadiusOfX*u*Sin[theta], 0} PlotTopSurface[RadiusOfA_, RadiusOfX_, Height_, t_, theta_, u_] := Module[{x, y}, x = RadiusOfA + (RadiusOfX - RadiusOfA) u; y = 2 Height*(1 - (2 RadiusOfX - 2 RadiusOfA)/(2 RadiusOfX - RadiusOfA - x)) + Height*(2 RadiusOfX - 2 RadiusOfA)/(2 RadiusOfX - RadiusOfA - x); {(x (1 - t) + RadiusOfA*t)*Cos[theta], (x (1 - t) + RadiusOfA*t)*Sin[theta], Height (1 - t) + y*t}] PlotSideSurface[RadiusOfA_, RadiusOfX_, Height_, t_, theta_, u_] := Module[{x, y}, y = Height*u; x = (2 RadiusOfX - RadiusOfA)*(1 - (2 Height/(2 Height - y))) + RadiusOfX (2 Height/(2 Height - y)); {(RadiusOfX (1 - t) + x*t)*Cos[theta], (RadiusOfX (1 - t) + x*t)*Sin[theta], y (1 - t)}] PlotRetract[RadiusOfA_, RadiusOfX_, Height_, t_] := ParametricPlot3D[ {PlotACylinder[RadiusOfA, Height, theta, u], PlotATop[RadiusOfA, Height, theta, u], PlotX[RadiusOfX, theta, u], PlotTopSurface[RadiusOfA, RadiusOfX, Height, t, theta, u], PlotSideSurface[RadiusOfA, RadiusOfX, Height, t, theta, u]}, {theta, 0, 2 Pi}, {u, 0, 1}, Mesh -> None, Axes -> None, Boxed -> False, PlotPoints -> 30, Lighting -> {{"Directional", White, {{1, 1, 1}, {0, 0, 0}}}}, PlotStyle -> {Gray, Gray, Gray, Directive[Blue, Opacity[0.5]], Directive[Blue, Opacity[0.5]]}] Export["animation.gif", Table[PlotRetract[1, 3, 4, Max[0, t]], {t, -0.1, 0.98, 0.02}], "DisplayDurations" -> {0.125}]