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

Does there exist an embedding $f: \mathbb{R} \rightarrow S^{2}$ 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 $f: \mathbb R \to S^1$ topological embedding and let $R=f(\mathbb R)$ . If $R$ is closed in $S^1$ $R$ 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

Answer

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