cianfa72
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- A 2-sphere as set can't be homeomorphic to the euclidean plane with any topology assigned to it
Suppose there was a bijection ##\varphi## between the 2-sphere ##M## and the euclidean plane ##\mathbb R^2##.
Then one could endow ##M## with the initial topology from ##\mathbb R^2## through ##\varphi## turning it into an homeomorphism (this topology on ##M## would be different from the subset topology inherited from sitting in ##\mathbb R^3## with its standard topology).
Since there is no bijection ##\varphi: M \to \mathbb R^2## as sets, then the 2-sphere as set cannot be endowed with any topology such that it would result homeomorphic to the euclidean plane, right ?
Then one could endow ##M## with the initial topology from ##\mathbb R^2## through ##\varphi## turning it into an homeomorphism (this topology on ##M## would be different from the subset topology inherited from sitting in ##\mathbb R^3## with its standard topology).
Since there is no bijection ##\varphi: M \to \mathbb R^2## as sets, then the 2-sphere as set cannot be endowed with any topology such that it would result homeomorphic to the euclidean plane, right ?
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