Sufficient to check intersection of sub base elements with a dense set in compact open topology

Question

I am reading a book about duality and there was this following claim: If I have a compact group G* (dual group of G, and G is discrete) with the compact open topology, then for any A, a subset of G* which contains the identity, it is sufficient to check non-empty intersection with any sub base elements of G*, in order to determine that A is dense in G. (i.e. any element of the form P(k,v) where k is compact in G and v open in T) My question is why it is sufficient to check only sub base elements and not all open sets in G*?