# Topological spaces with the Freese--Nation property II

## TL;DR

This paper investigates topological spaces with the FNS and π-FNS properties, establishing their connections to openly generated and skeletally generated spaces, respectively, through properties of bases and generation methods.

## Contribution

It demonstrates that compact spaces with the FNS property are openly generated, and those with the π-FNS property are skeletally generated, linking these properties to space generation concepts.

## Key findings

- Compact FNS spaces are openly generated.
- π-FNS spaces are skeletally generated.
- Connections between FNS properties and space generation methods.

## Abstract

The aim of this paper is to study the class of spaces with the FNS property and $\pi-\FNS$ property. We shown that compact spaces with the FNS property for some base consisting of cozero-sets are openly generated spaces and spaces with the $\pi-\FNS$ property are skeletally generated spaces.