= Sphere with its equator collapsed
{title2=$S^2/S^1\cong S^2\vee S^2$}
Collapse the equator of a <sphere> $S^2$ to a single <equivalence class> and leave every other point in its own class. The two closed hemispheres are <disks> sharing that equator. A disk with its boundary collapsed is a sphere, so the two hemisphere quotients are spheres sharing just one point: their <wedge sum>. An explicit map sends height $t\in[0,1]$ and longitude $\theta$ on the upper hemisphere to $(1-\cos\pi t,\sin\pi t\cos\theta,\sin\pi t\sin\theta)$, and sends the lower hemisphere to the sphere centred at $(-1,0,0)$ by reversing the first coordinate. The equator goes to the common origin and the poles go to $(\pm2,0,0)$. This map is continuous and has precisely the collapsed equator as its only nonsingleton fibre. Its induced map is a <homeomorphism> by the <compact-to-Hausdorff continuous bijection theorem>.
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