I 2-sphere with any topology can't be homeomorphic to the plane

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 ?
 
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What makes you think there is no bijection ##\mathbb R^2 \to \mathbb S^2##? Both sets are ##\aleph_1##.
 
Orodruin said:
What makes you think there is no bijection ##\mathbb R^2 \to \mathbb S^2##? Both sets are ##\aleph_1##.
Ah..and how can one define such a bijection? For example stereographic projection from North pole maps bijectively all points on ##\mathbb S^2## to the plane but the North pole itself.
 
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Start with the stereographic projection ##S_p##. You now have a bijection from the sphere except a single point ##p## to ##\mathbb R^2##. Now make room for ##p## in ##\mathbb S^2 \setminus p##: Let ##s = (x_i)_{i \geq 0}## be any countably infinite sequence of distinct points in ##\mathbb S^2## such that ##x_0 = p## and define
##B: \mathbb S^2 \to \mathbb S^2 \setminus p## such that
$$
B(x) = \begin{cases} x_{i+1},& \quad x = x_i \in s\\
x, &\quad x\notin s \end{cases}
$$
which is clearly a bijection. The composition ##S_p \circ B## is now a bijection from ##\mathbb S^2## to ##\mathbb R^2##.
 
Orodruin said:
The composition ##S_p \circ B## is now a bijection from ##\mathbb S^2## to ##\mathbb R^2##.
Very interesting! So by mean of ##S_p \circ B## one can assign ##\mathbb S^2## the initial topology from ##\mathbb R^2##. With this topology ##\mathbb S^2## is homeomorphic to the Euclidean plane.

Therefore the claim that the 2-sphere ##\mathbb S ^2## is not homeomorphic to the Euclidean plane, assumes ##\mathbb S^2## endowed with the subspace topology from sitting in Euclidean ##\mathbb R^3##.
 
cianfa72 said:
Very interesting! So by mean of ##S_p \circ B## one can assign ##\mathbb S^2## the initial topology from ##\mathbb R^2##. With this topology ##\mathbb S^2## is homeomorphic to the Euclidean plane.

Therefore the claim that the 2-sphere ##\mathbb S ^2## is not homeomorphic to the Euclidean plane, assumes ##\mathbb S^2## endowed with the subspace topology from sitting in Euclidean ##\mathbb R^3##.
I mean, if you don’t want to endow ##\mathbb S^2## with the subspace topology, one might question the point of letting it be that particular subspace at all …
 

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