I'll first state the question as it appears:

Let and suppose that for every is countable. Prove that is countable.

To me: , doesn't make sense, I'm interpreting this as one specific n-tuple, so I don't understand how an n-tuple can or cannot be countable. So I interpreted the question as . This may be wrong right away so if someone can explain to me why it should be the other way that would help.

But here's my attempt at a solution based on my interpretation:

I'm trying to use the Lindelof covering theorem which states that: Suppose . If is an open covering of , then there is a countable subcollection that is also a cover of . But I'm not able to make much progress with that method. And as of now I can't think of another way of showing it.

Any tips, or hints would be appreciated. Thanks in advance