Let $X$ be the quotient space of $S^1 \times S^1$ by the equivalence relation $(z,w) \sim (\bar z, \bar w)$. prove $X$ is homeomorphic to $S^2$.
2026-03-26 21:10:19.1774559419
Homeomorphism of quotient space of $S^1 \times S^1$
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Your starting point looks good.
Interpret $S^1$ as the interval $[0, 1]$, with $0$ and $1$ glued. The complex conjugation $z \leftrightarrow \overline{z}$ then becomes $a\leftrightarrow 1 - a$.
Now as you said, we can write $S^1 \times S^1$ as the unit square $[0, 1]\times[0, 1]$ with both opposite sides glued. The equivalence relation translates to $(a, b) \sim (1 - a, 1 - b)$.
The rest is best explained by a picture: