A bump in the road in elementary topology

TL;DR

The paper highlights a subtle, often overlooked issue in the definitions of relative and weak topologies in elementary topology, which could impact understanding and application of these concepts.

Contribution

It identifies a subtle difficulty in the standard definitions of relative and weak topologies that has been generally unnoticed.

Findings

Unnoticed difficulty in defining relative topology

Potential implications for topology education and research

Highlights need for careful consideration of topology definitions

Abstract

We observe a subtle and apparently generally unnoticed difficulty with the definition of the relative topology on a subset of a topological space, and with the weak topology defined by a function.