# Quotient space of a triangle?

by iLoveTopology
Tags: quotient, space, triangle
 P: 743 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.