Understanding Quotient Spaces of Triangles

[iLoveTopology]

Hello, I have been given a homework problem and I don't want any help on solving the problem, (I'm not even going to post the problem - I want to figure it out myself), I only want to understand what the problem is asking. (That's why I've posted in this section rather than the homework section. This is just a general concept question.)Essentially, there are pictures of two triangles (I'm going to draw a pic and link to the pic)

http://i.imgur.com/VKpet.jpg

The question begins "consider the following identification spaces of a triangle"

So I know that identification space means quotient space. But I have a difficult time understand what this quotient space here is supposed to be. So this is my understanding of a quotient space:

You have some topological space, X, and an equivalence relation ~ on it. The quotient space, A, is then the set of equivalence classes on X. Now, if you have a surjective mapping from X on to A and you define sets in A as being open if their preimage in X is open, then this defines a topology on A, and we call this the quotient topology.So what is the equivalence relation ~ in these diagrams? Does it mean that the interior points are all just equivalent to themselves (they don't change) but that the points on each edge are an equivalence class? Also, I understand how a triangle or polygon in general can have an orientation. For example, the triangle on the left has an orientation but the one on the right does not appear to be orientable. But what does this orientation have to do with the equivalence relation ~?

 

[Kreizhn]

In the diagram, the interior points are not identified with anything, only the boundaries of the triangle are identified. The arrows on the boundary tell you how to make the identifications. The triangle can be difficult to visualize, so perhaps it would be best to use a square as an example.

Consider this diagram, which shows how to typical torus may be viewed as a quotient space (in particular, \mathbb R^2/\mathbb Z^2). Naively, the identification corresponds to gluing the edges represented by the same arrows in a way such that the arrows line up. In this case, the orientation of the arrows is very important. Take a look at this figure, which is still a square but now one of the arrows has switched orientation. This is an fact the Klein bottle. Similarly, this diagram shows that if we only identify two sides of the square (with the "same" orientation) then we get the Moebius band. If we switch one of those arrows, we would get a cylinder.

We needn't stick to squares! The projective line can be seen as the circle with antipodal points identified.

Now do a similar thing with your triangles. The issue here is that, unlike the examples that I gave above, it may be quite difficult to visualize what the corresponding quotient space. To get an idea of what is happening, it may be useful to try drawing some paths within the triangles, and see what happens when they cross the identified boundaries.

 

[iLoveTopology]

hey kreizhn, thank you very much this is a detailed response and helps a lot. I kept getting hung up on the fact that there were no edge labelings that would tell me what to "glue together" as in the examples of the square for the torous and klein bottle. But I guess I was completely overlooking the fact that because there are only three edges in the triangle, if all the edges have orientations, they all kind of mesh together in some way so you don't have to label the edges to indicate which ones go together. Yes, I am finding this difficult to visualize but your idea of drawing paths and seeing where they go is very helpful. Thank you!