Cellularity of first countable spaces |
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Institution: | Department of Mathematics, York University, North York, Canada M3J 1P3 |
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Abstract: | In this paper, we find subspaces of the Pixley-Roy space on the irrationals which are - 1.(1) a first countable ccc space which does not have a σ-linked base,
- 2.(2) for each n>1, a first countable space which has a σ-n-linked base but which does not have a (σ−n+1)-linked base and
- 3.(3) a first countable space which has, for each n>1, a σ-n-linked base but which does not have a σ-centered base.
It is consistent with ¬CH that (1) and (2) have cardinality ℵ1. (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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