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## Main Question or Discussion Point

My notes claim that the square root loop [tex] \sigma : S^1 \longrightarrow RP^1 ; z=\cos 2\pi t +i\sin 2\pi t \longrightarrow [\cos \pi t, \sin \pi t] [/tex] is a homeomorphism, where [x,y] is an equivalence class given by the antipodal equivalence relation on the circle. However, this map doesn't even seem to be a bijection. The proof simply says "by construction". Can anyone help?