I am having difficulties grasping the consequences of this theorem, would really appreciate a little enlightenment.(adsbygoogle = window.adsbygoogle || []).push({});

A:Well, the statement of the theorem is clear, that Every closed Surface is homeomorphic to:

1) a sphere,

2) the connected sum of g tori

3) or the connected sum of g Projective plane.

B:Closed surface means Every closed 2 dimensional manifold that is embeddable in [tex]R^n [/tex]. So [tex]S^3[/tex] or in fact [tex]S^n [/tex] is homeomorphic to one of the above.

C:Another point is that [tex]S^n \cong R^{n+1}[/tex]

So by transitivity we get [tex]R^4 \cong S^3 \cong S^2 \cong R^3 [/tex], which is false by the invariance of dimensions.

Given A,B and C i come to this conclusion which i know to be wrong, so can someone explain to me where i went wrong.

The only conscious leap i have made is B, which i haven't read from any book, but given the definition i don't see why it should be otherwise.

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# Classification Theorem of Surfaces

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