an:04032592
Zbl 0634.54015
Steprāns, Juris; Watson, Stephen
Cellularity of first countable spaces
EN
Topology Appl. 28, No. 2, 141-145 (1988).
0166-8641
1988
j
54D65 05C55
Pixley-Roy space; first countable ccc space; \(\sigma \)-centered base; complete subgraphs
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).