O ] 1 5 M ar 1 99 4 Cardinalities of topologies with small base ∗

Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T0). We prove that if T has a base of cardinality ≤ μ, λ ≤ μ < 2, λ strong limit of cofinality א0, then T has cardinality ≤ μ or ≥ 2. This is our main conclusion (21). 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 (for λ = א0 this fulfill a promise from [Sh 454]). The main result for the case λ = א0 was proved in [Sh 454]. ∗Partially supported by The Basic research Fund, Israeli Academy of Sciences. Publication no. 454A done 8/1991, 3-4/1993. I thank Andrzej Roslanowski for proofreading, pointing out gaps and rewriting a part more clearly.