Sh:454a

• Shelah, S. (1994). Cardinalities of topologies with small base. Ann. Pure Appl. Logic, 68(1), 95–113.
• 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].
• published version (19p)
Bib entry
@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}
}