= Solution
Every quotient class has a representative in $[0,1]$: the collapsed class has representatives $0,1$, and the others are the points of $(0,1)$. Thus $p|_{[0,1]}$ is a continuous surjection from a <compact interval> onto $X_A$. It follows that \b[$X_A$ is <compact>], despite the noncompact original space.
Define the continuous map
$$
F(x)=\begin{cases}e^{2\pi ix},&0\leq x\leq1,\\1,&x\leq0\text{ or }x\geq1.\end{cases}
$$
The formulas agree at both junctions, and its fibres are exactly the required <equivalence classes>. It induces a continuous bijection $X_A\to S^1$, which is a <homeomorphism> by compactness and the <Hausdorff condition> on the <circle>.
$$
\boxed{\mathbb R/\bigl(({-\infty},0]\cup[1,\infty)\bigr)\cong S^1.}
$$
The quotient notation means that the displayed entire subset becomes one point.
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