The hierarchy of Borel universal sets

A Σα0-subset U of the product X×Y is a Σα0-universal set of X parametrised by Y if every Σα0-set of X is of the form Uy={x: (x,y)∈U}, for some y∈Y. Let n∈ω and α∈ω1. If X is a compact space with a Σn0-universal set parametrised by Y, then for all m∈ω, w(X)⩽nw(Y), hd(Xm)⩽hd(Ym), hL(Xm)⩽hL(Ym) and hc(Xm)⩽hc(Ym). If X is a compact perfect space with a Σα0-universal parametrised by Y, then w(X)⩽nw(Y). The statements “every compact monotonically normal space with a Σα0-universal set parametrised by a second countable space is metrisable” and “every compact, first countable space with a Σα0-universal set parametrised by a second countable space is metrisable” are undecidable in ZFC. Relevant examples are presented.