 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 $X_A$ is compact, despite the noncompact original space.

Define the continuous mapThe 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.The quotient notation means that the displayed entire subset becomes one point.