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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).

##### MSC:
 54D65 Separability of topological spaces 05C55 Generalized Ramsey theory
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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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