Weakly countably determined spaces of high complexity

TL;DR

This paper explores the complexity of weakly countably determined spaces, showing they can surpass coanalytic complexity, but also characterizing coanalytic sets via specific families of closed sets.

Contribution

It demonstrates the existence of weakly countably determined spaces with higher complexity than coanalytic and characterizes coanalytic sets through cofinal adequate families.

Findings

Existence of weakly countably determined spaces of higher complexity than coanalytic.

Coanalytic sets characterized by cofinal adequate families of closed sets.

Banach spaces from these families have at most coanalytic complexity.

Abstract

We prove that there exist weakly countably determined spaces of complexity higher than coanalytic. On the other hand, we also show that coanalytic sets can be characterized by the existence of a cofinal adequate family of closed sets. Therefore the Banach spaces constructed by means of these families have at most coanalytic complexity.