# On some kinds of weakly sober spaces

## TL;DR

This paper investigates the properties of weakly sober spaces, a generalization of sober spaces, exploring which characteristics are preserved or lost in this broader context.

## Contribution

It analyzes the basic properties of weakly sober spaces introduced by Erné, highlighting similarities and differences with sober spaces.

## Key findings

- Some properties of sober spaces are preserved in weakly sober spaces.
- Certain properties of sober spaces do not hold in weakly sober spaces.

## Abstract

In \cite{E_2018}, Ern\'e relaxed the concept of sobriety in order to extend the theory of sober spaces and locally hypercompact spaces to situations where directed joins were missing, and introduced three kinds of non-sober spaces: cut spaces, weakly sober spaces, and quasisober spaces. In this paper, their basic properties are investigated. It is shown that some properties which are similar to that of sober spaces hold and others do not hold.