MHB Is $\Bbb R$ homeomorphic to a cartesian product?

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Here is this week's POTW:

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Prove that if $\Bbb R$ is homeomorphic to a cartesian product $A\times B$, then either $A$ or $B$ is a singleton.

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No one answered this week's problem. You can read my solution below.

Spoiler

Let $(a,b)$ be a point of $A \times B$ corresponding $0$ under a homeomorphism $\Bbb R \to A \times B$. Then there is a homeomorphism $\Bbb R\setminus \{0\} \to (A\times B) \setminus \{(a,b)\}$. Therefore, $(A \times B)\setminus\{(a,b)\}$ is disconnected. Since $A \times B$ is connected, then $A$ and $B$ are connected. So if neither $A$ nor $B$ is a singleton, then $(A\times B)\setminus\{(a,b)\}$ is connected. ($\rightarrow\leftarrow$)