Cellularity of first countable spaces.

*(English)*Zbl 0634.54015We 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).

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).

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\textit{J. Steprāns} and \textit{S. Watson}, Topology Appl. 28, No. 2, 141--145 (1988; Zbl 0634.54015)

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##### References:

[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 |

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