Sh:454a

• Shelah, S. (1994). Cardinalities of topologies with small base. Ann. Pure Appl. Logic, 68(1), 95–113. arXiv: math/9403219 DOI: 10.1016/0168-0072(94)90049-3 MR: 1278551
• Abstract:
  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 \leq \mu, \lambda\leq \mu< 2^\lambda, \lambda strong limit of cofinality \aleph_0, then T has cardinality \leq \mu or \geq 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].
• Version 1994-08-07_10 (26p) published version (19p)

@article{Sh:454a,
 author = {Shelah, Saharon},
 title = {{Cardinalities of topologies with small base}},
 journal = {Ann. Pure Appl. Logic},
 fjournal = {Annals of Pure and Applied Logic},
 volume = {68},
 number = {1},
 year = {1994},
 pages = {95--113},
 issn = {0168-0072},
 mrnumber = {1278551},
 mrclass = {54A25 (03E75)},
 doi = {10.1016/0168-0072(94)90049-3},
 note = {\href{https://arxiv.org/abs/math/9403219}{arXiv: math/9403219}},
 arxiv_number = {math/9403219}
}