How a Change to Topology's Union Axiom Affects Continuity

TL;DR

This paper explores how modifying one of the axioms defining a topology impacts the concept of continuity in mathematical spaces, highlighting the foundational importance of topological axioms.

Contribution

It investigates the effects of altering a topology's union axiom on the definition and properties of continuous functions.

Findings

Changing the union axiom modifies the class of open sets.

Altered axioms lead to different notions of continuity.

The study clarifies the foundational role of axioms in topology.

Abstract

The most general definition of a continuous function requires that the preimage of any open set be open. Thus, to discuss continuity in the abstract, it is necessary to first define a topology, which tells us which sets in a space are open. Such a topology is given by three axioms that describe how the open sets in a topology behave. In this paper we shall consider a change to one of these axioms and determine how this change affects the continuity of a function.