### Cardinalities of topologies with small base

by Shelah. [Sh:454a]

Annals Pure and Applied Logic, 1994

Let T be the family of open subsets of a topological
space (not necessarily Hausdorff or even T_0). We prove that
if T has a base of cardinality <= mu, lambda <=
mu < 2^lambda, lambda strong limit of cofinality aleph_0,
then T has cardinality <= mu or >= 2^lambda . This is
our main conclusion. First we prove it under some set theoretic
assumption, which is clear when lambda = mu ; 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
lambda = aleph_0 this fulfill a promise from [Sh 454]). The
main result for the case lambda = aleph_0 was proved in [Sh
454].

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