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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 [itex] (x,y) \sim (x', y') \Longleftrightarrow \text{ either } (x,y) = (x', y') \text{ or } x = 1  x' \text{ and } y  y' = \pm 1 [/itex]
Why do we need [itex](x,y) = (x', y') [/itex] in the relation? Indeed, why do we need [itex] y  y' = \pm 1 [/itex]?
Surely all we need is [itex] (x,y) \sim (x', y') \Longleftrightarrow x = 1  x' \text{ and } y  y' = 1 [/itex]
Can anyone explain how the relation [itex] (x,y) \sim (x', y') \Longleftrightarrow \text{ either } (x,y) = (x', y') \text{ or } x = 1  x' \text{ and } y  y' = \pm 1 [/itex] 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 [itex] (x,y) \sim (x', y') \Longleftrightarrow \text{ either } (x,y) = (x', y') \text{ or } x = 1  x' \text{ and } y  y' = \pm 1 [/itex]
Why do we need [itex](x,y) = (x', y') [/itex] in the relation? Indeed, why do we need [itex] y  y' = \pm 1 [/itex]?
Surely all we need is [itex] (x,y) \sim (x', y') \Longleftrightarrow x = 1  x' \text{ and } y  y' = 1 [/itex]
Can anyone explain how the relation [itex] (x,y) \sim (x', y') \Longleftrightarrow \text{ either } (x,y) = (x', y') \text{ or } x = 1  x' \text{ and } y  y' = \pm 1 [/itex] actually works to produce the Mobius Band?
Peter
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