# 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]. - 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} }