# Embedding $\mathbb{R}$ into $S^{2}$

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.

• The term "embedding" (in the question title, but not in the question body) usually means "$f$ is a homeomorphism onto its image". Just checking: Do you want that condition, or not? – Andrew D. Hwang Apr 9 '15 at 10:42
• Try to use the claim known as the Invariance of Domain. Consequently this fact $f$ must be surjective too. (if $f$ is a topological embedding) – Leonhardt von M Apr 9 '15 at 10:46
• @user86418 I want an embedding, yes, thanks. – lisyarus Apr 9 '15 at 10:48

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.