P-domination and Borel sets

TL;DR

This paper explores the concept of P-domination in topological spaces, linking it to Borel sets and providing new characterizations that extend previous results on K-analytic and Polish spaces.

Contribution

It introduces a modified definition of P-domination that characterizes Borel subsets of Polish spaces, expanding the understanding of domination concepts in topology.

Findings

P-domination characterizes Borel subsets of Polish spaces.

A small modification of P-domination links to Borel sets.

Extends previous results on K-analytic and Polish spaces.

Abstract

In recent years much attention has been enjoyed by topological spaces which are dominated by second countable spaces. The origin of the concept dates back to the 1979 paper of Talagrand in which it was shown that for a compact space X, Cp(X) is dominated by P, the set of irrationals, if and only if Cp(X) is K-analytic. Cascales extended this result to spaces X which are angelic and finally in 2005 Tkachuk proved that the Talagrand result is true for all Tychnoff spaces X. In recent years, the notion of P-domination has enjoyed attention independent of Cp(X). In particular, Cascales, Orihuela and Tkachuk proved that a Dieudonne complete space is K-analytic if and only if it is dominated by P. A notion related to P-domination is that of strong P- domination. Christensen had earlier shown that a second countable space is strongly P-dominated if and only if it is completely metrizable. We show that a very small modification of the definition of P-domination characterizes Borel subsets of Polish spaces.