I am reading Martin Crossley's book, Essential Topology.

I am at present studying Example 5.55 regarding the Mobius Band as a quotient topology.

Example 5.55 Is related to Examples 5.53 and 5.54. So I now present these Examples as follows:

I cannot follow the relation $\displaystyle (x,y) \sim (x', y') \Longleftrightarrow \text{ either } (x,y) = (x', y') \text{ or } x = 1 - x' \text{ and } y - y' = \pm 1 $

Why do we need $\displaystyle (x,y) = (x', y') $ in the relation? Indeed, why do we need $\displaystyle y - y' = \pm 1 $?

Surely all we need is $\displaystyle (x,y) \sim (x', y') \Longleftrightarrow x = 1 - x' \text{ and } y - y' = -1 $

Can anyone explain how the relation $\displaystyle (x,y) \sim (x', y') \Longleftrightarrow \text{ either } (x,y) = (x', y') \text{ or } x = 1 - x' \text{ and } y - y' = \pm 1 $ actually works to produce the Mobius Band?

Peter