Circle modulo reflection ($S^1/(z\sim \overline{z})\cong [0,1]$)

Identify points of the circle having the same real coordinate, equivalently points related by complex conjugation. The continuous surjection $h(z)=(1+\mathrm{Re}z)/2$ has exactly these equivalence classes as its fibers. It induces a continuous bijection from the quotient space to the closed interval. The quotient space is compact as a continuous image of a compact space, while the closed interval is Hausdorff, so the compact-to-Hausdorff continuous bijection theorem makes the induced map a homeomorphism. The two fixed points of the reflection become the two interval endpoints.