I have some questions in regards to how to show homeomorphism between a space $X$ with an equivalence relation $\sim$ defined on it, and its new quotient space $Y$
In the attached image taken from a playlist of lecture videos on Topology and Groups, 3.01 Quotient Topology shows four progressively complicated examples. The first one being the classic example of identifying two end points of a unit interval $I$ as equivalent, and any points within the unit interval gets map to itself. The resulting quotient space is a circle $S^1$. To show that this unit interval with the associated equivalence relation defined over it is indeed a circle requires one construct a homeomorphic mapping $q$ from $I/{\sim}$ to $S^1.$ The mapping usually given is a function in parametric form: $f(\theta)=(\cos(\theta), \sin(\theta)).$
The next example in the attached image is the example of collapsing the boundary $A=\partial D^2$ of a disk $D^2$ to a point, and hence the resulting quotient space is a sphere. To show that $X/A$ is homeomorphic to $S^2$, the homeomorphic mapping $q$ can be explicitly written out in parametric form.
However, for the next two examples, defining an equivalence relation on a torus with the resulting quotient space being the pinched torus, and defining an equivalence relation on an octagon with the resulting quotient space being the double torus. To show that these spaces with their equivalence relations defined over it, are respectively homeomorphic to the pinched torus and double torus. I have never seen that any function written out in parametric form be given. My question is, if a parametric function can not be explicitly constructed to show homeomorphism between $X/{\sim}$ and the resulting quotient space $Y$. What other mathematically rigorous methods can be used to do so. Thank you in advance.
I think the examples are only intended to give an intuitive illustration what the quotient spaces look like.
In the first two examples the spaces on the right side are easy to describe: They are the spheres $S^1$ and $S^2$, and it is not difficult to construct explicit homeomorphisms.
In the following two examples the spaces on the right side are not really properly defined. You certainly understand that they are suitable subsets of $\mathbb R^3$, but there is no precise definition. You can do that, but is would be very tedious to define them as concrete sets of points, and I believe it is not worth the effort. The difficulty is then to get a homeomorphism between an exactly defined quotient space and a "vaguely defined" object.
In the third example you may define the pinched torus as the quotient space on the left side, but then the object on the right side is not the pinched torus, and you still have to explictly describe it and to construct an explicit homeomorphism. Yes, it can be done, but see my above remarks.
In the fourth example you can show that the quotient space is a compact $2$-dimensional surface without boundary (in the abstract sense of manifold). These objects are well-know and can be classified (see https://en.wikipedia.org/wiki/Surface_(topology), https://en.wikipedia.org/wiki/Genus_g_surface). You can show that the quotient space is orientable with genus $2$, thus topologically a double torus. However, this is by no means trivial and does not give an explicit homeomorphism.