Also, with regards to an interior of a closure, is this interpretation:

If A is open then the interior of it's closure is A itself

Not necessarily. For example, the set has closure [0,2]. The point 1 is in the interior of the closure, but is not in the original set.
... if A is closed then the interior of it's closure is a subset of A.

Correct?

Next,

Part of the BCT as it's stated in the book I'm working from is: Let (X,d) be a complete metric space. Then if U1, U2, U3... are countably many dense

open subsets of X then the intersection of N Un's is dense in X.

I only have a small photocopied part of this book, and I have been introduced to "countable" from another book as meaning either finite or countably infinite. Is it true that some authors (including this one) use it only to mean countably infinite, and NOT finite? Otherwise, where does this break down - the rationals and irrationals are both dense in the reals, but their intersection is null, and the closure of the null set is null, which is not the reals... ?