Definition talk:Connected Sum

Shouldn't the union be a disjoint union (with the Definition:Disjoint Union (Set Theory))? I'm trying to think what would happen if we had two manifolds which contain some of the same points as sets (i.e. they have a nonempty intersection). For example, if $A^n = B^n = T^2$, where $T^2$ is the torus $\{\mathbb{x} \in \mathbb{R}^3 \mid (\sqrt{x_1^2 +x_2^2} - 1)^2 +x_3^2 = 1\}$, and $\alpha, \beta$ are the same embedding of the disk but traced in opposite directions, then $S = (A^n \setminus \alpha(({\Bbb D^n}^\circ)) \cup (B^n \setminus \beta(({\Bbb D^n}^\circ)) = (T^n \setminus \alpha(({\Bbb D^n}^\circ)) \cup (T^n \setminus \alpha(({\Bbb D^n}^\circ)) = (T^n \setminus \alpha(({\Bbb D^n}^\circ))$. That is, $S$ would be a torus with a hole punched in its surface. Then I think the quotient $S/\sim$ would just stitch the hole back together, so we would essentially be left with the original torus. But the usual result is that $T^2 \# T^2$ is homeomorphic to the two-holed torus? --plammens (talk) 09:12, 9 October 2021 (UTC)