# Locally ordered topological spaces

## TL;DR

This paper explores the properties and characterizations of locally ordered topological spaces, focusing on their separation axioms, connectedness, and compactness, and provides new insights into their structure and examples.

## Contribution

It introduces a comprehensive study of locally ordered spaces, filling the gap in understanding their topological properties and relationships with separation axioms.

## Key findings

- Characterization of connected, locally connected, and compact locally ordered Hausdorff spaces
- Analysis of the relationship between local orderability and separation axioms
- Presentation of interesting examples of locally ordered spaces

## Abstract

While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and separation axioms and give characterisation of all connected, locally connected or compact locally ordered Hausdorff spaces. A collection of interesting examples is also offered.