Embedding R\mathbb{R} into S2S^{2}

Does there exist an embedding f:RS2 with a closed image? I believe not, but I’m stuck with how to prove that.

It would be nice to hear several different proofs if my guess is true.

Answer

Let f:RS1 topological embedding and let R=f(R). If R is closed in S1 R is compact and f is a homeomorphism between a compact and a non compact topological space. This is a contradiction.

Attribution
Source : Link , Question Author : lisyarus , Answer Author : Leonhardt von M

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