# Absolute $F_{\sigma\delta}$ spaces

## TL;DR

This paper proves that certain topological spaces, including separable Banach spaces with the weak topology, are absolutely $F_{\sigma\delta}$, extending their properties across compactifications.

## Contribution

It establishes that hereditarily Lindelöf $F_{\sigma\delta}$ spaces are absolutely $F_{\sigma\delta}$, with implications for Banach spaces in weak topology.

## Key findings

- Hereditarily Lindelöf $F_{\sigma\delta}$ spaces are absolutely $F_{\sigma\delta}$.
- Separable Banach spaces with weak topology are absolutely $F_{\sigma\delta}$.
- Extension of $F_{\sigma\delta}$ property across compactifications.

## Abstract

We prove that hereditarily Lindel\"of space which is $F_{\sigma\delta}$ in some compactification is absolutely $F_{\sigma\delta}$. In particular, this implies that any separable Banach space is absolutely $F_{\sigma\delta}$ when equipped with the weak topology.