# Hatcher 2.1.1

Hatcher, Algebraic Topology, Chapter 2, Section 1

1. What familiar space is the quotient $\Delta$-complex of a 2-simplex $[v_0,v_1,v_2]$ obtained by identifying the edges $[v_0,v_1]$ and $[v_1,v_2]$, 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 $[v_0,v_2]$ to $v_1$, and rearranging. I (crudely) illustrate the process here. $\square$