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Let $\tau$ be the topology generated by half-open intervals of the form $[a,b)$ where $a$ is a rational number and $b$ is a real number. Let $C$ denote the space endowed with the previously described topology.

Prove/disprove: $C \times C$ is a Lindelof space.

How do you proceed: here $C$ is not equal to the Sorgenfrey line (because of the rational endpoint). Do we have to use Jones lemma like when showing $\mathbb{R}_{l} \times \mathbb{R}_{l}$ is not Lindelof?

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Hint: You can show that $C$ is second countable.

Update after the first 2 comments: Once you have that, what can you say about $C\times C$? What can you say about the relationship between second countability and Lindelöfness?

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    @Henno: Thank you for explaining.2011-02-06