Hello everyone. I have the following problem.

The quote says to consider that if were second countable that every open base must have a countable subset which is also an open base.Let with the usual topology for every and let . Prove that is not second countable.

But, I don't understand why this doesn't work.

Clearly we have that where is open in (in fact it's a subbasic open set). So, we have that is an open set in . If were second countable we would have, by Lindelof's theorem, that must have a countable subcover. But, that isn't the case here. Right?