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My question is:
Should the deleted point of the torus be the intersection point of the two circles? If yes why? and if no, why also?
2026-02-23 04:45:26.1771821926
A small discussion about Hatcher Q.1, chapter 0
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You must not remove the intersection $p$ of the circles $S_l$ (longitude circle) and $S_m$ (meridian circle) because in that case the complement $T \setminus \{p\}$ cannot retract to $S_l \cup S_m$ (a retract $A$ of a space $X$ must be a subset of $X$).
Hatcher's square is the product $I \times I$ with $I = [-1,1]$. He removes te origin $(0,0)$ which after identification of opposite sides does not lie on $S_l \cup S_m$.