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

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

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