Let T be the family of open subsets of a topological space . We prove that if T has a base of cardinality μ, λμ<2λ, λ strong limit of cofinality 0, then T has cardinality μ or 2λ. This is our main conclusion . In Theorem 2 we prove it under some set-theoretic assumption, which is clear when λ = μ; then we eliminate the assumption by a theorem on pcf from [Sh 460] motivated originally by this. Next we prove (...) that the simplest examples are the basic ones; they occur in every example . The main result for the case λ = 0 was proved in [Sh 454]. (shrink)