Theorem A1.2(4). Suppose x is in the intersection of the open sets S1,S2,...,Sn. The intersection is assumed to be non-empty. Then since each Si is open there exist ei>0, i=1,...n such that Bei(x) is in Si. Let e=min{e1,...,en}>0. Then Be(x) is in Bei(x) is in Si for all i. So Be(x) is in the intersection of the sets Si and we have the result.
To see what goes wrong with an infinite intersection, consider Si={x:(-1/i,1/i)} i=1,2,... The intersection is {0}, a closed set. We cannot find a minimum e>0 - this is the importance of the 'finite' bit. Hope this helps. The book by Berge is very good if you need more help on topology.