Hatcher, Algebraic Topology, Chapter 2, Section 1
1. What familiar space is the quotient -complex of a 2-simplex obtained by identifying the edges and , preserving the ordering of the vertices?
Proof. It’s a Möbius band.
This can be seen by considering the 2-simplex, cutting along the perpendicular bisector connecting to , and rearranging. I (crudely) illustrate the process here.