enter image description here 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.

  • $\begingroup$ How do you define the pinched torus and the double torus? To explicitly construct homeomorphisms, you have to answer this question before. $\endgroup$ – Paul Frost Jan 2 at 15:16
  • $\begingroup$ @PaulFrost the pinched torus is defined as $(S^{1}\times S^{1})/S^{1}\times \{1\}$, I don't know the notation for a double torus other than is some three dimensional surface with two holes in it.. You also raised a good question, what happens if one does not know how to even define the new quotient space. The point of showing homeomorphism between spaces in topology requires one to construct or write out an explicit function.. I would say virtually all texts in topology on the topic of quotient spaces, it never states how much mathematical details that requires. $\endgroup$ – Seth Mai Jan 2 at 20:52
  • $\begingroup$ If you define the pinched torus as you do, then everything is trivial. But you are right, details are frequently omitted. If you want to construct a homeomorphism, you must precisely define the spaces before. For the double torus one can in fact invoke that it is the essentially unique genus $2$ surface. But that is not at all elementary. $\endgroup$ – Paul Frost Jan 2 at 21:00
  • $\begingroup$ @PaulFrost I often see similar questions asked and the various commenter sometimes would say something along the following, if you know $X/\sim$ is compact and the targeted quotient space $Y$ is Hausdorff, then the following theorem homeomorphsim between compact and Hausdorff spaces can be invoked as a way to show that such a homeomorphic map $f$ can be constructed without actually having to do so. However, looking at that particular theorem, my impression is that one still requires to construct $f$ $\endgroup$ – Seth Mai Jan 2 at 21:10
  • $\begingroup$ The quoted theorem only says that if you are given a continuous bijection $f : A \to B$ from a compact $A$ to a Hausdorff $B$, then it is a homeomorphism. But it does not save constructing $f$. $\endgroup$ – Paul Frost Jan 3 at 13:33

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.

  • $\begingroup$ I have seen the following concepts in some topology texts like "connected sums", "Euler Characteristics", "wedge sums", "smashed products", "adjunction spaces." Does any of these concepts help with showing existence of homeomorphisms between a space with an equivalence relation defined over it and its quotient spaces? My impression of this particular topic of describing spaces is the following, in the beginning, one has to use a lot of efforts in the sense of having to construct explicit functions.... $\endgroup$ – Seth Mai Jan 3 at 20:23
  • $\begingroup$ ...later on, as one learns more advanced concepts when taking a course in either algebraic/geometric topology, one learns more elaborate tools which allows one to go about showing that such type of function exist even though constructing will be a difficult process. We don't have to look far to see how complicated it can be for giving explicit construction of homeomorphic functions. Take the case of the Klein bottle. Most topology texts would just give a verbal description of how an equivalence relation could be defined on a space to get this quotient space.... $\endgroup$ – Seth Mai Jan 3 at 20:27
  • $\begingroup$ ..but it rarely ask the reader to give an explicit homeomorphic mapping like a mapping in parametric form. The other issue is, if i am given a compact space with an equivalence relation defined on it, is there some sort of a guarantee even that regardless of the type equivalence relation being defined on it, the resulting quotient space will be Hausdorff. $\endgroup$ – Seth Mai Jan 3 at 20:30
  • $\begingroup$ All constructions and concepts that you mentioned may be helpful to understand the nature of a space. However, they will not really help you to visualize a quotient space. I think a true visualization is only possible if the quotient space has a homeomorphic "model" in $\mathbb R^2$ or $\mathbb R^3$. But then we end with the problem that we need to construct an explicit homeomorphism or to invoke deep theorems of (algebraic) topology. A quotient of a compact Hausdorff space is in general not Hausdorff. Take $[0,1]$ and identify $(0,1]$ to a point. The quotient space is not Hausdorff. $\endgroup$ – Paul Frost Jan 4 at 0:32
  • $\begingroup$ Thank you for the clarification with the example about how a quotient space can be compact but not hausdorff. Sorry for my late response. I have been feeling under the weather. Anyways, I just want to add the following example as an illustration on quotient space and homeomorphism. The example is from Singh's Elements of Topology.: "A Mobius band (or strip) is a surface in $\mathbb{R}^3$ generated by moving a line segment of finite length in such a way that its middle point glide along a circle, and the segment remains normal to the circle... " $\endgroup$ – Seth Mai Jan 7 at 2:24

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