So I'm asked to proof that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic. So far, I've been able to prove that $\mathbb{R}\backslash\{a\}$ and $\mathbb{R}^2\backslash\{b\}$ are not homeomorphic, for $a\in \mathbb{R}$ and $b\in \mathbb{R}^2$. But I don't know how to go on from here. Can anyone give me a hint?
General Topology – How to Prove $\mathbb{R}$ and $\mathbb{R}^2$ are Not Homeomorphic
general-topology
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Best Answer
Hint. If $f \colon \def\R{\mathbf R}\R \to \R^2$ were a homeomorphism, what does this imply for the restriction $f\colon \R \setminus \{a\} \to \R^2 \setminus\{f(a)\}$?