Cardinalities of topologies with small base

Saharon Shelah

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Cardinalities of topologies with small base

Saharon Shelah

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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].

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BibTeX RIS

Saharon Shelah. 1994-03-15. Cardinalities of topologies with small base. https://arxiv.org/abs/math/9403219

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