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

2006-08-25

arxiv.org/abs/math/0608636v1

in AMS, DIMACS: Series in Discrete Mathematics and Theoretical Computer Sciences, ed. by S. Thomas 58 (2002), 45-63

Mathematics

Logic

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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