On $D$-spaces and Discrete Families of Sets

Details On $D$-spaces and Discrete Families of Sets On $D$-spaces and Discrete Families of Sets

2008-11-07

arxiv.org/abs/0811.1165v1

On $D$-spaces and Discrete Families of Sets, in AMS, DIMACS: Series in Discrete Mathematics and Theoretical Computer Sciences,

Mathematics

Logic

and old paper, the previous version on arxiv only contained the latex macros

Scientific paper

We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using the concept of discrete families. We note the connection between non-reflection arguments involving discrete families and the well known question of the existence of families allowing partial transversals without having a transversal themselves, and use it to give non-trivial instances of the incompactness phenomenon in the context of discretisations.

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