Hatcher Exercise 2.1.1


Take a 2-simplex and form the quotient space given by identifying the faces (edges) and preserving the ordering of vertices.

The resulting quotient space will be a compact surface with boundary, and is homeomorphic to a Mobius strip. We can see this with the classification of compact surfaces (with boundary):

After identifying the edges, there is one vertex, two edges, and one face, giving , then observing that the the surface is non-orientable (because we have a pair of edges identified in the ‘same direction’ around the polygon), so the surface is either or (the Mobius strip), and since there is at least one boundary component (along the unidentified edge, for instance), it must be .