zbMATH — the first resource for mathematics

Cellularity of first countable spaces. (English) Zbl 0634.54015
We find subspaces of the Pixley-Roy space on the irrationals which are (1) a first countable ccc space which does not have a \(\sigma\)-linked base, (2) for each \(n>1\), a first countable space which has a \(\sigma\)-n- linked base but which does not have a \((\sigma -n+1)\)-linked base and (3) a first countable space which has, for each \(n>1\), a \(\sigma\)-n-linked base but which does not have a \(\sigma\)-centered base.
It is consistent with \(\neg Ch\) that (1) and (2) have cardinality \(\aleph_ 1\). (3) is constructed from a graph G on the continuum c which is not the union of countably many complete subgraphs but has no uncountable pairwise incompatible family of finite complete subgraphs (complete subgraphs A and B are compatible if there is a complete subgraph C which contains A and B).

54D65 Separability of topological spaces
05C55 Generalized Ramsey theory
Full Text: DOI
[1] M. Bell, Two Boolean algebras with extreme cellular and compactness properties. Canad. J. Math, to appear. · Zbl 0519.06012
[2] Hajnal, A.; Juhasz, I., A consequence of Martin’s axiom, Indag. math., 33, 457-463, (1971) · Zbl 0302.54005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.