= Solution
{c}
Let $i:A\hookrightarrow X$ be the <inclusion map> and $r:X\to A$ the <retraction>. Since $X$ is <contractible>, choose a <contraction of a topological space>
$$
H:X\times I\longrightarrow X,\qquad H(x,0)=x,\quad H(x,1)=x_0.
$$
The <function composition>
$$
K:A\times I\longrightarrow A,\qquad K(a,t)=r(H(i(a),t))
$$
is a <continuous map>. Its endpoints satisfy
$$
K(a,0)=ri(a)=a,\qquad K(a,1)=r(x_0).
$$
Consequently $K$ is a <homotopy> from the <identity map> of $A$ to a <constant function>. This proves
$$
\boxed{A\text{ is contractible}.}
$$
The contracting point of $X$ need not belong to $A$: applying the <retraction> sends it to the contracting point $r(x_0)$ of $A$. This is the <retract of a contractible space> principle.
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