Cardinalities of topologies with small base
Logic
2016-09-06 v1 General Topology
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 <= 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].
Cite
@article{arxiv.math/9403219,
title = {Cardinalities of topologies with small base},
author = {Saharon Shelah},
journal= {arXiv preprint arXiv:math/9403219},
year = {2016}
}