# The $\pi$-property of a Banach space along a filter

## TL;DR

This paper studies the complexity of the class of separable Banach spaces with the $ ext{pi}$-property defined via filters, showing it is analytically complex ($ ext{Sigma}^1_3$ or $ ext{Sigma}^1_2$) depending on the filter's properties.

## Contribution

It establishes the descriptive set-theoretic complexity of the class of Banach spaces with the $ ext{pi}$-property based on the nature of the filter, extending understanding of their analyticity.

## Key findings

- The class is $ ext{Sigma}^1_3$ when the filter is analytic.
- If the filter is countably generated, the class is $ ext{Sigma}^1_2$.
- The results depend on the filter's properties and the topology on the space of closed subspaces.

## Abstract

We examine the analyticity of the class of separable Banach spaces possessing the $\pi$-property, defined in terms of convergence along a filter. Our results establish that this class is $\Sigma^1_3$ whenever the underlying filter is analytic (as a subset of the Cantor set $\Delta$). Furthermore, we demonstrate that if the filter is countably generated, the class of such spaces is $\Sigma^1_2$ with respect to any admissible Polish topology on the family of closed subspaces of $C(\Delta)$.