# How can one visualize topological quotients or develop intuition for handling them?

This is a very open-ended question. I regret that -- I would like to be able to make it more precise, but I don't know how. I would appreciate comments on how to improve this question.

I had my first course in topology a couple of years ago and did later very little about the subject. I received a good grade then and thought I wouldn't have much trouble during the Topology II course. Unfortunately, this is very far from the truth. I'm struggling to understand almost every sentence that is said there. And one of the things that cause me the most trouble is quotient spaces.

I know the definition of course and I know that taking quotients means "gluing equivalent points together". The problem is that it doesn't give me much insight into how a given quotient space behaves. I find myself staring with my jaw dropped at people juggling with different interpretations of the same space as quotients of different spaces. When the real projective plane was discussed, I asked how I can visualize it. I was told to "simply take a sphere and glue antipodal points together." Well, fine, I understand that this is a well-defined operation on topological spaces, but the problem is that I can't visualize such a thing at all. It's impossible to do this to a sphere in the world I live in.

I've studied mathematics long enough to understand that I won't be able to visualize everything. But in this case, other people seem to have no problems with this. I would like that too. Could you please help me with it? I would appreciate answers explaining particular examples of quotient spaces that cannot be embedded in $\mathbb R^3,$ but also any other kind of answer that you think might help me overcome my problem.

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Perhaps this should be made community wiki... –  user38268 Mar 28 '12 at 22:43
@ymar When you edit the question there should be a tiny check box at the bottom right of the text field. –  Matt N. Mar 28 '12 at 22:53
Made community wiki by request. @Matt: note that users cannot make questions community wiki by themselves anymore. –  Zev Chonoles Mar 28 '12 at 22:59
Why should this be a community wiki question? –  t.b. Mar 28 '12 at 23:02
@ymar: I also think that this is fine as a regular question. While it may not have a single correct answer, it isn't a poll-style question, and I think we should still reward useful answers to it. I'd be happy to undo the CW if you change your mind. –  Zev Chonoles Mar 28 '12 at 23:05

## 1 Answer

There is a bit of case-by-case here, but when the quotient is "nice" enough, you can often use a fundamental domain to visualise it.

You probably know how to construct a torus as a quotient of the plane by identifying points that are translations apart. You can visualise this torus also by taking the parallelogram generated by the two translations and thinking like pac-man. That is, following a point inside the region and if it hits one of the sides, it pops back on the other side since the two sides have been "identified".

This also works for the projective plane quotient you talked about. Since the antipodal points are being identified, the northern hemisphere of the sphere is a fundamental region, that is, it contains exactly one point in each equivalence class. So you can see the projective plane as a hemisphere, where when you move a point to the equator it pops back to the opposite side. Since we're thinking only topologically here, you could also "flatten" this hemisphere to a disk, so the projective plane is like a disk where when you reach the border you reappear at the antipodal point, much like the torus picture.

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This is very helpful, thank you very much! –  user23211 Mar 29 '12 at 23:11