The Hierarchy of Borel Universal Sets

by

Paul M. Gartside and Joseph Tzan Hang Lo

Proceedings of the Third Galway Topology Colloquium (1999)

A \Sigma_\alpha⁰-subset U of the product X×Y is a \Sigma_\alpha⁰-universal set of X parametrised by Y if every \Sigma_\alpha⁰-set of X is of the form U^y = { x: (x,y) in U}, for some y in Y. Let n in \omega and \alpha in \omega₁. If X is a compact space with a\Sigma_n⁰-universal set parametrised by Y, then for all m in \omega,w(X) <=nw(Y), hd(X^m) <=hd(Y^m), hL(X^m) <=hL(Y^m) and hc(X^m) <=hc(Y^m). If X is a perfect compact space with a \Sigma_\alpha⁰-universal set parametrised by Y, then w(X) <= nw(Y). There is an example of a space that is not second countable, but with a G_\delta-universal set parametrised by the Cantor set, 2^\omega. Assuming b = \omega₁, there is a locally compact strong S-space with a G_\delta-universal set parametrised by 2^\omega. Assuming CH, there is an L-space with a G_\delta-universal set parametrised by 2^\omega. Assuming b = \omega₁, there is a compact strong S-space with a G_\delta-universal parametrised by a strong S-space. If there exist Q-sets or under CH, there is a compact, first countable non-metrisable space with a \Sigma_\omega⁰-universal set parametrised by 2^\omega. The statements "every compact monotonically normal space with a \Sigma_\alpha⁰-universal set parametrised by a second countable space is metrisable" and "every compact, first countable space with a \Sigma_\alpha⁰-universal set parametrised by a second countable space is metrisable" are undecidable in ZFC.

Keywords: Borel hierarchy, Borel universals, cardinal invariants, S and L spaces, compact spaces
MCS: 54H05, 54D30 Secondary 54D65, 54E35, 54D15

This article was published. Gartside, Paul M.; Lo, Joseph T.H. The hierarchy of Borel universal sets. Topology and its Applications, Vol: 119, Issue: 2, pp. 117-129; PII S0166-8641(01)00070-0

Date received: September 24, 1999.