# Separation axioms as lifting properties

## TL;DR

This paper demonstrates that many classical separation axioms in topology can be uniformly expressed using the language of category theory as lifting properties, simplifying their conceptual understanding.

## Contribution

It introduces a categorical framework for expressing separation axioms as lifting properties, unifying and simplifying their descriptions.

## Key findings

- Separation axioms can be characterized by lifting properties.
- Category theory provides a uniform language for topology axioms.
- Finite spaces and the real line are sufficient for these characterizations.

## Abstract

We observe that many of the separation axioms of topology (including $T_0-T_4$) can be expressed concisely and uniformly in terms of category theory as lifting properties (in the sense of Quillen model categories) with respect to (usually open) continuous maps of finite spaces (involving up to 4 points) and the real line.