On $D$-spaces and Discrete Families of Sets

TL;DR

This paper explores reflection theorems for $D$-spaces, a class of topological spaces, using set-theoretic assumptions, and connects combinatorial properties of discrete families to incompactness phenomena.

Contribution

It establishes new reflection theorems for $D$-spaces under set-theoretic assumptions and links discrete family combinatorics to incompactness in topology.

Findings

Reflection theorems for $D$-spaces under forcing axioms and large cardinal assumptions.

Connection between discrete families and incompactness phenomena.

Examples of non-reflection involving discrete families.

Abstract

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.