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Let $\mathbb{R}_{\ell}$ the Sorgenfrey line and let $\mathbb{Q}$ endowed with usual topology.

I have two questions:

1) Is there a continuous surjective map $f: \mathbb{R}_{\ell} \rightarrow \mathbb{Q}$?

2) Is there a continuous surjective map $f: \mathbb{R}_{\ell} \rightarrow \mathbb{R}$ where $\mathbb{R}$ has the usual topology?

Not sure where to start. Can you please help?

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    @user10: You could make your answer community wiki if you just don't want to gain reputation for answering your own question.2011-07-05

1 Answers 1

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Let $f: \mathbb{R}_{\ell} \rightarrow \mathbb{Z}$ be the map given by the floor function, then $f$ is surjective and continuous. Now $\mathbb{Z}$ and $\mathbb{Q}$ are both countable so there is a bijection $g: \mathbb{Z} \rightarrow \mathbb{Q}$. Since $\mathbb{Z}$ is discrete this map is cts and surjective so $g \circ f$ is the desired map.