In every book they say: gluing points together in a topological space is done via the quotient topology. Then they provide some examples, but they don't try to explain WHY it is true for any kind of example we can think of:
- Considering $[0,1]/\sim$ with the quotient topology identifying $0$ with $1$ leads to a homeomorphism to the circle. That's also what our imagination says!
- Another examples is identifying the left and right site of a rectangle in the same direction, which leads to $[0,1]^2/\sim$ with the quotient topology, which is homeomorphic to a cylinder. That's also what our imagination says!
- Again, another example is considering the closed disk identifying the boundary $\mathbb{D}/\sim$ and using the quotient topology to get a homeomorphism to the sphere $\mathbb{S}^2$. That's also what our imagination says!
My question:
If we consider an object $X\subset \mathbb{R}^3$ and want to glue points together, our imagination in our head(!) leads to an object $Y\subset \mathbb{R}^3$ (just glue the points in your head together). But the books always say: Consider $X/\sim$ and take the quotient topology on it.
What is a heuristic argument, that $X/\sim$ is homeomorphic to $Y$? I want a "why it should be true quotient topology leads really to gluing:…" and not just "your three examples provide evidence, that is good enough and should be true for all other cases…".
EDIT: Everyone of you stuck at the point, that $Y$ may not be in $\mathbb{R}^3$. That is not the point of my question. Just restrict your attention to examples, where $Y$ is in $\mathbb{R}^3$.
I don't want to take the definition for granted. I want to understand them and convince myself that's how we should do it. If you do research, you also have to find the right definition to capture the behavior you want. That may be the most difficult part in mathematical research.
MAIN QUESTION:
If you never heard of the quotient topology (say we developed the theory of topological spaces one week ago), but you want to formalize the concept of gluing. You have a topological space $(X,\tau)$. Now you have an equivalence relation $\sim$ on $X$ and want to get a topological space $(X/\sim, ?)$.
How would you come up with the right topology on the quotient set $X/\sim$ to catch the behavior of gluing in a mathematical precise manner?
Best Answer
Your question is too vague to give a precise answer to. After all, I don't understand what your Y is or why you think it was assembled by gluing or why it still sits in $\Bbb R^3$. Until you can tell me what gluing is I can't tell you why that's what quotients are.
Still, here is the result that unifies all of your three examples. I leave the proof as an exercise to you.
Whatever your space Y is, when you tell me that you are imagining gluing points of X together, you are presumably telling me a map from X to Y. That's what the above is. Then the result says that Y is precisely the quotient space obtained by identifying the points in X that f does.
In your first example, f traces out the circle counterclockwise. The second example is similar. In your third example, f stretches a rubber disc over the 2-sphere with the entire boundary of that disc getting mapped to the bottom point.