The Möbius band and the Klein's bottle:

We describe the Möbius band and Klein's bottle as quotient spaces of $I^2$ via identifications which are described as follows. Each point in the interior of $I^2$ forms an equivalence class in itself. That is to say a point in the interior of $I^2$ is not identified with any other point. Points on the boundary are identified according to the following scheme:

1. Möbius band: On the part of the boundary $(\{0\}\times[0, 1])\cup (\{1\}\times [0, 1])$, the pair of points $(0, y)$ and $(1, 1-y)$ are identified for each $y$ with $0 \leq y \leq 1$. Points on the remaining part of the boundary namely
   $$(0, 1)\times \{0\}\cup (0, 1)\times \{1\} \eqno(4.3)$$
   are left as they are. That is to say the equivalence class of each of the points (4.3) is a singleton.

   Figure: Möbius Band

   [width=.4]GKSBook/fig7/fig7.eps
2. Klein's bottle: As in the case of the Möbius band, for each $0 \leq y \leq 1$, the pair of points $(0, y)$ and $(1, 1-y)$ are identified. However also for each $0 \leq x \leq 1$, the pair of points $(x, 0)$ and $(x, 1)$ are identified.

   Figure: Klein's Bottle

   [width=.4]GKSBook/fig8/fig8.eps