Is it necessary to specify "how we glue something" in topology Consider a hollow sphere $S$, upon which we have two points on opposite end $O, O'$. It is possible to "glue" these two points in two different ways.
One way, is to consider someone pinching on opposite ends until they connect, forming effectively a torus, with an inner hole of size 0.
the other way, is to pull at these opposite ends, stretching out the sphere, and having the two pulled ends meet outside, along $O - O'$. 
So now's my question, are these two objects homeomorphic? What would be the most effective way to reason that out?
And if they aren't, then how do topologists decide "how to glue" points, given that the choice of how they attach points, affects the resulting shape.
 A: Yes they are homeomorphic. The issue with which way to pull is a side issue. When you glue points together, you are not thinking of the sphere as being embedded in space, and in fact there is only one way to glue the two points together if you think of this as an abstract space. It turns out that if you do embed the sphere with points identified in the two ways you describe, the two embeddings are actually ambient isotopic (Edit: in the $3$-sphere), but that's a different question than homeomorphism.
A: Yes, these two spaces are homeomorphic. The easiest way to see that is to understand how you take quotients in topology: you first forget about the topology and take the quotient by the equivalence relation as a set, and then you put the final topology on the quotient set with respect to the quotient map. Thus, all that matters is which points are identified. 
More abstractly, this is how you perform any sort of limit (product, fibre product etc.) or colimit (quotient, disjoint union etc.) construction for topological spaces: take the set theoretic limit (resp. colimit) and then put the initial (resp. final) topology with respect to the "inclusion" (resp. "projection") maps of the limit (resp. colimit). 
