When is a Lindelöf topological space separable?

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Let X be a topological space (neither metrizable nor second countable ).

  1. If X is Lindelöf, when X is separable?
  2. If X is separable, when X is Lindelöf?
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  1. See my answer to the same question.

  2. I'll try to look for an answer when I'll have time.