Point-set topology as diagram chasing computations: Lifting property as negation

TL;DR

This paper reveals that many fundamental topological properties can be understood as lifting properties, framing them as diagram chasing computations, which offers a new perspective on their logical structure and potential implications for AI.

Contribution

It introduces the idea that key topological definitions are essentially lifting properties relative to simple counterexamples, connecting topology with diagram chasing computations.

Findings

Many topological properties are lifting properties.

Some arguments in topology resemble diagram chasing computations.

Potential links between topology, logic, and AI cognition are discussed.

Abstract

We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms $T_0$ and $T_1$ in topology, having dense image, induced (pullback) topology, and every real-valued function being bounded (on a connected domain). We also offer a couple of brief speculations on cognitive and AI aspects of this observation, particularly that in point-set topology some arguments read as diagram chasing computations with finite preorders.