# $G_\delta$-refinements

## TL;DR

This paper investigates how certain topological properties are preserved under $G_\delta$-refinements in specific classes of scattered spaces, addressing open questions about tightness in $G_\delta$-refinements of $\sigma$-products.

## Contribution

It establishes preservation results for metacompactness, paralindel"ofness, metalindel"ofness, and linear lindel"ofness in $ ext{SP}$-scattered spaces under $G_\delta$-refinements, and explores related generalizations.

## Key findings

- Preservation of several covering properties in $ ext{SP}$-scattered spaces.
- Analysis of $G_\delta$-refinements in $ ext{N}$-scattered and $ ext{ω}$-scattered spaces.
- Addressing the question of tightness in $G_\delta$-refinements of $\sigma$-products.

## Abstract

In this work we deal with the preservation by $G_\delta$-refinements. We prove that for $\mathrm{SP}$-scattered spaces the metacompactness, paralindel\"ofness, metalindel\"ofness and linear lindel\"ofness are preserved by $G_\delta$-refinements. In this context we also consider some other generalizations of discrete spaces like $\omega$-scattered and $N$-scattered. In the final part of this paper we look at a question of Juh\'asz, Soukup, Szentmikl\'ossy and Weiss concerning the tighness of the $G_\delta$-refinement of a $\sigma$-product.