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Intuitively speaking, one space is homemorphic to another if one can be deformed continuously to another without tearing and gluing. It is more or less easy to convince the students that a square is homeomorphic to a triangle.

How about a strip with two full twists and a standard annulus then? How can I illustrate to my students that it can be continuously deformed to a standard annulus? I failed to find an easy way to convince the students that we can "untwist" the two twists using continuous deformations without tearing and gluing.

enter image description here

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    $\begingroup$ Not all homeomorphisms can be achieved via ambient isotopy in R^3 ("continuous deformation"), and teaching (college) students this gives them some misconceptions. For example, it's impossible to turn a knot into an unknot in this way, but both are homeomorphic. Similarly the Mobius strip can have some "handedness" depending on its embedding in R^3 which cannot be resolved via ambient isotopy as well (e.g. this image physicsdetective.com/wp-content/uploads/2018/08/…) $\endgroup$ – Opal E Jun 24 '19 at 18:24
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    $\begingroup$ It is closer to the truth to say that you are permitted to tear and glue, as long as points which were near each other before the tearing stay near each other after the gluing. $\endgroup$ – Steven Gubkin Jun 24 '19 at 19:21
  • $\begingroup$ @OpalE, thanks for your comment! So the common intuitive understanding that "homeomorphism is continuous deformation without tearing and gluing" is incorrect? $\endgroup$ – Zuriel Jun 25 '19 at 18:14
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    $\begingroup$ Yes, homeomorphism is more general than the intuitive understanding. "continuous deformation without tearing and gluing" can be thought of as something called an "isotopy" which is more specific than a homeomorphism. A homeomorphism is a continuous one-to-one map whose inverse is also continuous. Continuous just means that points that start near each other, end near each other. Steven's description that "you are permitted to tear and glue, as long as the points which were near each other before you tear stay near each other" is a better description of what a homeomorphism is, in general. $\endgroup$ – Opal E Jun 25 '19 at 18:20
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First of all, let me echo all the comments -- the key point here is that these surfaces are homeomorphic, but this homeomorphism cannot be realized by an isotopy. This is an important distinction! It is probably worth taking some time to explain the issue. In particular, students should understand that topological spaces don't naturally come embedded in $\mathbb{R}^n$ and that homeomorphism is a property of the abstract spaces.

I think I would draw this picture on the black board and ask the students to complete the bottom diagram.

enter image description here

I haven't done this particular example in class, but I do use this sort of pictures for cylinders, Mobius strips, tori, Klein bottles and projective planes all the time, so it would tie into that familiarity.

I can imagine a fun activity here. Pass out the three types of strips to the students and have them triangulate them by drawing with pen. Have them carefully record which vertices are joined by edges and which three cycles are filled with triangles. Then pass that list of combinatorial data to another team and see if that team can determine which simplicial complex is which strip. If this works for you, let me know how it goes!

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  • $\begingroup$ As someone who isn't 100% well versed on this, this answer made the intuition very clear! I've seen these drawings before for cylinder/mobius strip/torus etc. so getting the students to physically do the last one should help them realize "oh...the edges will line up the exact same way as the cylinder did...so they must be the same!" $\endgroup$ – ruferd Jul 10 '19 at 12:26
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As suggested, I will turn my comment into an answer. Perhaps you could provide each student with a rubberband (or something similar) and provide different challenges for them by showing different shapes and asking which ones are possible and which ones aren't. You could even color the sides of the rubberbands different colors in case the different orientation matters (for something like a mobius strip).

This idea obviously doesn't create any rigorous proof, but it does provide the intuition/visual confirmation for what is/isn't going to be homeomorphic.

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    $\begingroup$ As per Opal E's comment, this is problematic, in that it is not possible to transform a doubly-twisted strip to an annulus in $\mathbb{R}^3$ without some tearing and gluing. Not all homeomorphisms are continuous deformations in a particular ambient space. $\endgroup$ – Xander Henderson Jun 24 '19 at 23:45
  • $\begingroup$ I'm not going to lie, I'm not completely sure what all that means. It has been a while since I had Topology afterall..But that's why I initially made this remark as a comment and not an answer, I was afraid that my limited knowledge on the subject would not be 100% correct. $\endgroup$ – ruferd Jun 25 '19 at 12:23
  • $\begingroup$ Ah, I think I see what my mis-understanding was. I thought that this was just a rubberband that had been twisted around itself twice. I am starting to see that it is a rubberband that was cut, one end twisted around twice, and then glued back together (some sort of 'double mobius strip' if you will). If that is actually the case, then I see that this answer is not that great. And now I am very interested in seeing some real intuition is for allowing a homeomorphism to cut and re-glue the shape. $\endgroup$ – ruferd Jun 25 '19 at 17:38

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